Structural Engineering & Statics
where the load goes
Connections

Making a moment cross a gap

A moment is not a thing that can be handed across a joint. It has to be turned into a pair of forces, carried separately, and reassembled — and the whole design is a question of how the tension is shared between bolt rows that are not equally able to take it.

Assumes The force the bolt never saw applied and Neither pinned nor rigid, which is every real connection.

18 min read 8 figures The load must go somewhereGeometry beats material

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.

How a moment crosses a gapA moment end plate with three bolt rows. The moment is carried as a couple: tension in the rows, compression through bearing at the bottom flange. The plastic distribution reaches 172.8 kN·m and the elastic one 142.29 kN·m, a factor of 1.21 — and the compression at the bottom flange is 540 kN either way, which is the check that gets forgotten because it is not a bolt.180 kNrow 1 · 420 mm180 kNrow 2 · 340 mm180 kNrow 3 · 200 mm540 kN compressionPlastic distribution — M = 172.8 kN·mthe other distribution is drawn faintly: 1.21× between them
Fig. 1 A moment end plate with three bolt rows. Tension in the rows near the top flange, compression through bearing at the bottom flange, and a lever arm between them. The plastic distribution shown lets every row reach its own limit and delivers 172.8 kN·m; the elastic distribution drawn faintly behind delivers 142.3. The compression at the bottom is 540 kN either way.

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 ∑Fihi\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 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.

How a moment crosses a gapA moment end plate with three bolt rows. The moment is carried as a couple: tension in the rows, compression through bearing at the bottom flange. The plastic distribution reaches 172.8 kN·m and the elastic one 142.29 kN·m, a factor of 1.21 — and the compression at the bottom flange is 411.43 kN either way, which is the check that gets forgotten because it is not a bolt.180 kNrow 1 · 420 mm145.71 kNrow 2 · 340 mm85.71 kNrow 3 · 200 mm411.43 kN compressionElastic distribution — M = 142.29 kN·mthe other distribution is drawn faintly: 1.21× between them
Fig. 2 The elastic distribution drawn as the governing one, with the plastic behind it. The top row is at its limit and the bottom row is at 48% of its own, because the arithmetic requires it to be proportional to a lever arm rather than to what it can carry. The bottom row is a bolt row that has been installed and is not being used.

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.

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.

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 hh above that centre extends by hθh\theta, so — if the rows are identical springs — its force is proportional to hh. Sum the moments: M=∑Fihi=c∑hi2M = \sum F_i h_i = c\sum h_i^2, and the constant cc 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 hθh\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 hh 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 ∑Fihi\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 z2z^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.

How a moment crosses a gapA moment end plate with three bolt rows. The moment is carried as a couple: tension in the rows, compression through bearing at the bottom flange. The plastic distribution reaches 223.2 kN·m and the elastic one 195.9 kN·m, a factor of 1.14 — and the compression at the bottom flange is 540 kN either way, which is the check that gets forgotten because it is not a bolt.180 kNrow 1 · 480 mm180 kNrow 2 · 420 mm180 kNrow 3 · 340 mm540 kN compressionPlastic distribution — M = 223.2 kN·mthe other distribution is drawn faintly: 1.14× between them
Fig. 3 The same three pairs of bolts with the lowest one moved above the top flange rather than below the middle. The moment goes from 172.8 to 223.2 kN·m — 29% more capacity, no more steel, and the compression at the bottom flange is unchanged at 540 kN because the tension total did not move. Only the arms did.
Prying action in a tee stubA tee stub pulled by its web with 100 kN per bolt. The 20 mm flange is in the one-hinge regime, so the prying force at the flange tip is 50.63 kN and the bolt carries 150.63 kN — 1.51 times what was applied. The flange stops prying entirely at 26.97 mm thick, and collapses on its own at 110 kN.100 kN appliedbolt 150.63 kNprying 50.63 kNm = 45n = 40flange 20 mm · one-hingebolt force is 1.51 times the applied load
Fig. 4 What each of those rows actually is. A bolt row in an end plate is a tee stub, and the row’s capacity is the tee stub’s — with prying, with three regimes, and with a thickness that removes prying entirely. There are two of them per row, one in the end plate and one in the column flange, and the row takes the lesser.
Prying against flange thicknessThe ratio of bolt force to applied force, for a tee stub carrying 100 kN per bolt, as the flange thickness varies. Prying disappears above 26.97 mm and the flange has become a mechanism below 19.13 mm, where the shaded region begins and the bolt has stopped being the thing that decides.05101520253011.21.41.6flange thickness, mmbolt force ÷ applied forceno prying above 26.97 mmflange is a mechanism
Fig. 5 And the same tee stub as a design curve, which is how the plate thickness gets chosen. Above 26.97 mm the row’s bolts carry exactly what the couple assigns them; below it they carry more, and the shaded region is where the plate has become a mechanism and the row’s capacity has stopped being a bolt capacity at all. The distribution question above assumed each row could reach its limit — this is the curve that says which limit that is.
How a moment crosses a gapA moment end plate with three bolt rows. The moment is carried as a couple: tension in the rows, compression through bearing at the bottom flange. The plastic distribution reaches 147.6 kN·m and the elastic one 94.86 kN·m, a factor of 1.56 — and the compression at the bottom flange is 274.29 kN either way, which is the check that gets forgotten because it is not a bolt.120 kNrow 1 · 420 mm97.14 kNrow 2 · 340 mm57.14 kNrow 3 · 200 mm274.29 kN compressionElastic distribution — M = 94.86 kN·mthe other distribution is drawn faintly: 1.56× between them
Fig. 6 The case the choice of distribution decides. The top row is weaker than the other two, and the elastic distribution scales everything to whichever row is most utilised relative to its lever arm — which is that one. The connection delivers 94.9 kN·m elastically against 147.6 plastically: a factor of 1.556, and over a third of the capacity lost to one row’s limit.

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.

A joint is springs in seriesThe five components of an end-plate joint, with each bar the flexibility it contributes. The column flange in bending is 39.62% of the total on its own, and doubling its stiffness raises the joint's by a factor of 1.25 — while doubling the stiffest component buys 1.05. The joint's rotational stiffness is 25227.71 kN·m per radian.flexibility contributed by each componentthey add, so the softest dominates — Sj = 25227.71 kN·m/radwhat doubling it buyscolumn web in shear21.89%×1.12column web in compression11.55%×1.06column flange in bending39.62%×1.25end plate in bending18.09%×1.1bolts in tension8.85%×1.05
Fig. 7 The same connection’s other question. Strength asks which component reaches its limit first and takes the minimum; stiffness asks how much each one deforms and adds the flexibilities. The decomposition is identical and the two answers name different components, which is why a joint can be stiff and weak or flexible and strong.
A base plate, and when the bolts start workingA 500 × 400 mm plate carrying 600 kN and 90 kN·m, so the resultant sits 150 mm from the centre against a kern of 83.33 mm. The plate is in partial contact: bearing over 300 mm at a peak of 10 N/mm², with the holding-down bolts carrying 0 kN. The plate lifts at 50 kN·m and crushes at 120 kN·m, and the bolts are not needed until 150 kN·m.600 kN90 kN·mresultant at e = 150middle thirdbolt carries nothingPartial contactbearing over 300 mm at 10 N/mm² · 50% of 20
Fig. 8 The same couple where one half of it has nothing to bolt to. A base plate carries its moment as tension in holding-down bolts and compression into grout — except that while a compression-only distribution exists, the bolts carry nothing at all and the whole couple is delivered by moving the bearing block. Same decomposition, and one of the two halves has gone missing.

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=∑FihiM = \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 hih_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.

Where this goes next

100 kN appliedbolt 150.63 kNprying 50.63 kNm = 45n = 40flange 20 mm · one-hingebolt force is 1.51 times the applied load The force the bolt never saw applied Pull a tee stub with a hundred kilonewtons and its bolt carries a hundred and fifty. The extra comes from the flange bending and pressing its own edge against the thing it is bolted to, and no free body of the connection as a point contains it. flexibility contributed by each componentthey add, so the softest dominates — Sj = 25227.71 kN·m/radwhat doubling it buyscolumn web in shear21.89%×1.12column web in compression11.55%×1.06column flange in bending39.62%×1.25end plate in bending18.09%×1.1bolts in tension8.85%×1.05 A joint made of springs in series A connection's stiffness is not a property of the connection. It is the series combination of the flexibilities of everything the force passes through — so the softest component decides, and stiffening any of the others changes almost nothing. neutral axiscompressiontensionI = 29.97 × 10⁶Z = 299.7 × 10³peak stress 200.2σ = M y ÷ I, at every height Bending is a pair of forces, pushing and pulling A bending moment is not a mysterious twisting. It is a push near the top of a section and a pull near the bottom, separated by a lever arm — a couple, made out of stress. 00.0050.010.0150.020.0250.030.0350.040.0450.05050100150200rotation, radiansmoment, kN·mrigid abovepinned belowweb cleats — pinnedflush end plate — semi-rigidextended end plate — semi-rigid Neither pinned nor rigid, which is every real connection Frame analysis offers two options for a joint and reality supplies a continuum between them. Worse, the boundaries are not properties of the connection at all — the same end plate is rigid on a short stiff beam and semi-rigid on a long slender one. 600 kN90 kN·mresultant at e = 150middle thirdbolt carries nothingPartial contactbearing over 300 mm at 10 N/mm² · 50% of 20 Where the structure meets the ground, and when the bolts start working Push a base plate off its middle third and it lifts off the foundation. The holding-down bolts then carry exactly nothing, and go on carrying nothing until the plate has crushed the concrete underneath it.

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Column webConnectionDuctilityEnd plateLever armMoment connectionPlastic distributionPrying