Making a moment cross a gap

A beam frames into a column and has to deliver a moment. Not a shear, which is a force and can simply be pushed across; a moment, which is not a force at all.

The only way a moment crosses a gap is as a couple: a tension somewhere, an equal compression somewhere else, separated by a lever arm. That decomposition is the whole of moment connection design, and every difficulty in it comes from the two halves of the couple being carried by completely different mechanisms.

The couple, and where its two halves go #

This is bending is a pair of forces applied to a joint rather than to a section, and the structure of the argument is identical: a moment is $\sum F_i h_i$, the forces sum to zero, and the lever arms do the work.

What is different is that the two halves are carried by unrelated things.

The tension goes into bolts. Each row of bolts pulls on the end plate, which pulls on the column flange, and every one of those is a tee stub with prying. So each row’s capacity is the least of four components: the end plate in bending, the column flange in bending, the bolts in tension, and the column web in tension.

The compression goes nowhere near a bolt. It is delivered by the beam’s bottom flange bearing directly against the end plate, which bears against the column flange. Its capacity is set by the beam flange in compression, the column web in compression, and the welds at the beam flange.

Nothing connects the two mechanisms except that they have to be equal.

The compression nobody checks #

That last sentence hides the commonest omission in moment connection design, and the figure puts a number on it.

Three rows at 180 kN each is 540 kN of tension. Equilibrium requires 540 kN of compression, and it arrives concentrated at the bottom flange over a contact patch a few tens of millimetres deep.

Five hundred and forty kilonewtons into an unstiffened column web, over about 100 mm of its depth. That is a column web crushing problem, or a column web buckling one if the web is slender — and neither of them is a bolt calculation, appears on a bolt schedule, or is what anybody means by “the connection”.

It is also the reason column web stiffeners exist opposite the compression flange, and why they are so often the item that makes a moment connection expensive. The bolts are cheap; the plate opposite them is not.

The comparison worth making is what the third row bought on each side of the couple.

That is the arithmetic behind an instruction that otherwise sounds like fussiness: a row low in the plate is a row added for shear, and adding it for moment loads the column web harder than it loads the bolts.

The general form is worth carrying: a couple has two halves and the analysis names only one of them. A designer who has sized the bolt rows has done half the connection, and the half that is left is a member check on the column.

There is a reason the compression half is the one that disappears, and it is linguistic rather than structural. The connection is called a bolted connection, its drawing is a bolt layout, its schedule is a list of bolts, and its checks are bolt checks. Nothing in any of that names the flange bearing at the bottom, because bearing needs no component and appears on no schedule. The half of the couple that is carried by a fastener is the half that gets an entry in the paperwork, and the half carried by two pieces of steel touching is the half that does not.

Two distributions, and the difference is not accuracy #

Given a set of rows at different lever arms, how much does each carry?

The elastic distribution makes each row’s force proportional to its lever arm, which is what a linear analysis of a rigid plate rotating about the compression centre gives. Scale it so that the most utilised row is at its limit and the rest follow: 180, 145.7 and 85.7 kN for rows at 420, 340 and 200 mm. Moment 142.3 kN·m.

The plastic distribution lets every row reach its own limit: 180, 180 and 180. Moment 172.8 kN·m.

A factor of 1.214 between them, for exactly the same steel. That is a larger gain than most things in this field, and it is free — if it is available.

What decides which one is available #

The plastic distribution requires the top row to reach its capacity and then keep carrying it while the rows below deform enough to reach theirs. That is a ductility requirement, and whether it is met depends on which of that row’s four components governs.

If the row is limited by the end plate in bending or the column flange in bending, the governing component is a plate forming a plastic mechanism, and a plate at its mechanism has a great deal of rotation available. The plastic distribution is available.

If the row is limited by the bolts in tension, it is not. A bolt at its tensile capacity is a fraction of a millimetre from fracture, and asking it to hold that load while the plate below deforms several millimetres is asking it to fracture.

So the design rule is precise and it inverts the usual instinct: the plastic distribution is available when the plate governs and not when the bolt does, and the way to make it available is to use a thinner plate or a stronger bolt — deliberately weakening the ductile component so that it, rather than the brittle one, is the limit.

Which of the three regimes a given row sits in is a question about one dimension, and it is the plate’s thickness rather than anything about the bolt.

That is the same preference the bearing essay and the component method both arrived at from different directions: make the ductile mechanism the weakest. Here it is worth 21% of the connection’s moment capacity — and the figure above says that choosing a plate a few millimetres thinner is how it is bought.

When the top row is the weak one #

The gain from plastic distribution is not a constant, and it grows as the rows become unequal.

Take the same three rows with the top one weakened — 120 kN instead of 180, which is what happens when the top row is close to the beam flange and its tee stub is stiffer and shorter. Now:

• elastic distribution: 94.9 kN·m
• plastic distribution: 147.6 kN·m
• gain: 1.556

The elastic distribution has collapsed, because it is scaled by whichever row is most utilised relative to its lever arm, and a weak row far out is exactly that. Every other row is then dragged down with it, and the connection loses over a third of its capacity to one row’s limit.

That is the case where the choice of distribution matters most and it is a common geometry. It is also a good illustration of why the elastic distribution is not merely conservative: it is conservative by an amount that depends on how unequal the rows are, which is to say by an amount nobody can estimate without doing both calculations.

Why the elastic distribution has that shape #

It is worth deriving the elastic distribution rather than asserting it, because the derivation says exactly what assumption is being made and therefore where it stops being right.

Assume the end plate is rigid and the joint rotates about the compression centre. Then a row at height $h$ above that centre extends by $h\theta$, so — if the rows are identical springs — its force is proportional to $h$. Sum the moments: $M = \sum F_i h_i = c\sum h_i^2$, and the constant $c$ is fixed by requiring the most utilised row to be at its limit.

Two assumptions in that, and both are false in ways this field has already met.

The plate is not rigid. It is the tee stub of the previous essay, and its flexibility is 40% of the joint’s total. A flexible plate lets the far rows extend less than $h\theta$ and the near ones more, flattening the distribution towards the plastic one.

The rows are not identical springs. A row near the beam flange has a shorter, stiffer tee stub than one in the middle of the plate, so it is both stiffer and stronger. Proportionality to $h$ assumes away exactly the variation that makes the weak-top-row case above so punishing.

So the elastic distribution is an idealisation of the same kind as a pinned joint or a rigid one, with the same property: it is exact at a limit nobody occupies. What makes it useful is not accuracy but that it is a lower bound — any distribution in equilibrium with the applied moment and within every component’s capacity is safe, by the same lower-bound theorem that lets a masonry arch be checked with any admissible thrust line. Elastic and plastic are two admissible distributions; plastic is the larger; both are safe if the ductility is there.

Where the rows should go #

The moment is $\sum F_i h_i$, so a row’s contribution is its force times its lever arm — and the lever arm is entirely a detailing choice.

Which produces the design instinct that dominates moment connections: get the top row as far from the compression flange as possible. Extending the end plate above the beam’s top flange, and putting a row outside the section, adds a row at the largest lever arm the geometry offers. That single row contributes more than an interior row at two-thirds of the distance, and it does so at both strength and stiffness — the $z^2$ in the stiffness formula rewards it quadratically.

It is the same argument as material far from the neutral axis, and it is now appearing for the sixth time on this site. The connections field has not introduced a new principle; it has introduced new places for the old one to apply.

The rows near the bottom, conversely, contribute almost nothing. A row at 200 mm against a top row at 420 carries less than half the moment per kilonewton, and in the elastic distribution it is not even at its capacity. Rows are added there for shear rather than for moment, which is a sensible division of labour and worth stating explicitly because the drawing does not distinguish them.

The arithmetic of that division is worth one line. In the elastic distribution the bottom row of the joint above carries 85.7 kN at a lever arm of 200 mm — 17.1 kN·m, or 12% of the connection’s moment. The same two bolts moved outside the top flange would sit at perhaps 480 mm and carry their full 180 kN, contributing 86.4 kN·m. Five times the moment from the same pair of bolts, and the only thing that changed is where they were drawn.

The same connection, with the moment the other way round #

Every figure here has the tension at the top and the compression at the bottom, which is a beam hogging over a support. Reverse the moment — wind on the frame, an uplift case, a pattern load that sags the span — and the couple turns over: compression at the top flange, tension at the bottom.

The connection does not turn over with it, and the advice of the previous section is what makes that expensive.

Getting the top row as far above the compression centre as possible produced the 223.2 kN·m arrangement, with a row sitting above the beam’s top flange. Reverse the moment and the compression centre moves to that top flange. The row that had the longest lever arm in the whole connection is now on the wrong side of the compression centre entirely, contributing nothing or worse; the rows that were at 420 and 340 mm are now at roughly nothing and 80 mm. The lever arms have collapsed, and with them the capacity.

So an extended end plate extended in one direction only is a strongly asymmetric connection: excellent in hogging, poor in sagging, and its drawing gives no indication of which. That is fine for a beam over a support in a braced frame, where the moment never reverses. It is not fine anywhere the load case list contains wind, and it is a real trap in a moment frame, where every joint sees moment in both senses by design.

The remedy is the obvious one and it is why the detail exists: extend the plate both ways, above the top flange and below the bottom one, with a row outside each. That restores the symmetry, roughly doubles the plate, and is the reason a moment-frame connection looks so much heavier than a moment connection in a braced frame carrying the same moment.

Which adds a qualification to the field’s recurring rule. Lever arm is still the cheapest capacity there is — but a lever arm is measured from the compression centre, and the compression centre moves when the load reverses. A detail optimised for the arms of one load case has been optimised against the arms of another.

The column web panel, which is the third mechanism #

There is a component that belongs to neither half of the couple and can govern the whole joint, and it is invisible on the connection drawing because it is inside the column.

Consider the column web in the region between the beam’s flanges. The tension arrives at the top of that panel and the compression at the bottom, both horizontally, in opposite directions. That is a shear applied to the panel, of magnitude equal to the couple’s force — 540 kN for the joint above.

A column web is a thin plate, and 540 kN of shear over a panel a few hundred millimetres square is a substantial demand. It can yield, and it can buckle, and either one lets the joint rotate without any bolt or plate reaching its own limit.

The panel is the reason a moment connection at an internal column is often easier than at an external one. With beams on both sides delivering moments in the same sense, their panel shears partly cancel — a downstand on one side pushes the panel one way and the other pushes it back. With a beam on one side only, nothing cancels.

It is also, from the component method’s list, the second-largest contributor to the joint’s flexibility at 21.9%. So the column web panel is simultaneously a strength check, a stiffness component and a possible failure mode, and it does not appear anywhere in the couple that the moment was decomposed into.

Why the shear is the easy part #

A moment connection carries shear as well as moment, and it is worth saying briefly why that half is uncontroversial.

The shear is carried by the bolts, or by friction, or by a seating — and it can be assigned to the bottom rows, which are contributing least to the moment. The two demands are therefore largely separable in practice, and the interaction between tension and shear in a bolt, while real, is not usually what governs.

That separability is a small mercy and it is why the essay has been about moment throughout. The awkwardness in a moment connection is entirely in the couple.

Two things this essay has not asked about the same connection are worth naming before it closes, because both use the same decomposition and neither gives the same answer.

The first is stiffness. Strength asks which component reaches its limit first and takes the minimum; stiffness asks how much each one deforms and adds the flexibilities. The list of components is identical and the two answers name different members of it, which is why a joint can be stiff and weak, or flexible and strong, without anything being inconsistent.

The second is the case where one half of the couple has nothing to bolt to. A column base carries its moment as tension in holding-down bolts and compression into grout — except that while the resultant stays inside the plate there is a compression-only distribution available, the bolts carry nothing at all, and the whole couple is delivered by moving the bearing block. Same decomposition, and one of its two halves is missing until the moment is large enough to lift the plate.

What to take from it #

A moment crosses a gap only as a couple, and the two halves are carried by unrelated mechanisms. Bolts and plates in tension; bearing and column web in compression.

The compression half is a member check on the column, and it is the one that gets missed. 540 kN into an unstiffened web over 100 mm, arising from a bolt calculation that never mentioned it.

Elastic and plastic distributions differ by 21% here and by 56% when the top row is weak. Which is available is decided by whether that row is limited by a plate or by a bolt — and detailing so that the plate governs is worth more than making anything stronger.

And the lever arm is the design variable, as it has been in every field on this site. Moving one pair of bolts from below the middle to above the top flange is worth 29% of the connection’s moment capacity — five times as much moment from the same two bolts — and costs a strip of plate.

There is a closing observation that belongs to the field rather than to the connection. Every result in this essay is an arithmetic consequence of $M = \sum F_i h_i$, which is the same equation the very first essay in the sections field opened with. What the connections field added was not a new mechanics but a new place where the $h_i$ are chosen by a detailer rather than fixed by a rolling mill. A section’s lever arms are set when the beam is specified; a connection’s are set when somebody decides where to put the bolts, and that decision is usually made to suit a spanner.