The moment the beam left behind

A frame model is a set of lines meeting at points. A beam ends where its centreline crosses a column’s centreline, and the reaction it delivers is applied there.

The building is not built that way. The beam ends about a hundred millimetres short of the column face, sitting on a cleat or bolted to an end plate, and the reaction it delivers arrives on that cleat. The distance between the two — the face of the column, plus a bit, plus half the column’s depth — is an eccentricity, and a force applied at an eccentricity is a force plus a couple.

Nothing about the force changes when it is moved: it is the same 80 kN, delivered by the same beam, and the moment it leaves behind follows the arm exactly. That is the whole of what an eccentric reaction does, and it is why the subject is arithmetic rather than analysis.

Which free body produced the number #

Take the column, cut above and below the connection, and put on it everything that touches it. Among those things is the beam’s reaction $R$, applied at a point a distance $e$ from the column’s axis.

Now move that force to the axis, which is the standard transformation: a force may be moved anywhere at the price of a couple. The column is then carrying $R$ on its axis plus a couple $Re$, and the two systems are equivalent for every equilibrium equation anybody can write about the column.

That couple has to be resisted, and the only things available to resist it are the column lengths above and below the joint. It is not carried by the connection — the connection is a pin and transfers no moment across itself — which is the sentence that makes the whole subject confusing. The joint has no moment capacity and it applies a moment to the column, because the moment does not pass through the joint. It is created by where the joint is, which is a fact about geometry rather than about what the connection can carry.

Eighty kilonewtons and a hundred millimetres #

The numbers are small individually and worth stating so their scale is clear. A beam reaction of 80 kN on a 203 mm deep column, with the connection assumed to act 100 mm out from the face, has an eccentricity of $203/2 + 100 = 201$ mm and delivers 16 kNm.

Sixteen kilonewton-metres is not much for a column carrying 800 kN. But it arrives at every floor, it is combined with an axial force that is at its maximum in the same load case, and it is applied about the column’s minor axis just as often as its major one — because beams frame into columns from both directions and the minor axis is where a column has least to give.

Set that beside the previous figure and the only thing that has changed is the arm. The 250 mm case gives 20 kNm and the 201 mm case 16.08, and neither number is available anywhere in the beam analysis that produced the 80 kN, because a reaction computed from a free body of the beam has a magnitude and a direction and no position on the support at all. The information the column needs is exactly the information the beam calculation never produced.

The cancellation, and where it fails #

The saving grace is that most columns get beams from both sides.

An internal column with equal beams framing in from left and right receives two eccentric reactions with opposite lever arms. Their couples subtract, and if the two reactions are equal the net moment on the column is exactly zero. That is why a column in the middle of a regular floor plate is a genuinely axially loaded member, and why the whole effect can be ignored on a great many projects.

It fails in four places, and each of them is a common arrangement rather than an unusual one.

A perimeter column has a beam on one side only. It gets the whole $Re$ with nothing to subtract.

A change of span — a 9 m bay meeting a 6 m bay — gets the difference of the two reactions times the eccentricity, which is most of what one side alone would give.

A pattern load removes the imposed load from one side. The two reactions were equal under full load and are not under the arrangement that governs, and it is the same argument that makes the envelope not a state of the structure.

A change of beam depth moves one connection’s eccentricity relative to the other, so even equal reactions do not cancel.

Of those four, the third is the one that reaches every column in a building rather than a few. Imposed load is by definition able to be absent, so every internal column in a frame has an arrangement in which one of its beams is loaded and the other is not — and the arrangement that maximises the eccentric moment on the column is not the one that maximises the beam moments, so it is a separate case that a run made for the beams will not contain.

How the couple divides #

Once the couple exists, the question is how much of it goes up and how much goes down.

The answer is the ordinary one for any out-of-balance moment at a joint: it is shared in proportion to $4EI/L$. Two column lengths of the same section and the same storey height take half each. A ground-floor column with a 5 m lower length and a 3.5 m upper length takes 41% below and 59% above.

The place this matters most is the top storey, where there is no column above, and the bottom, where the base may be a pin. In both cases one branch has no stiffness and the entire couple goes into the other one — the top-storey column length carries the whole of the roof beam’s eccentric moment, which is why the top length of a column is often not the lightest section in the stack.

Notice that the eccentricity argument does not care which of those curves a connection sits on. A joint with genuine rotational stiffness also transfers a moment through itself, which is a second and separate effect — neither pinned nor rigid is the essay about that one. The eccentric moment exists even for a connection with no stiffness at all.

What it costs the column #

A column carrying axial force and moment is checked on an interaction, and the shape of that interaction decides whether 16 kNm is a nuisance or a design case.

At low axial ratios the section has moment capacity to spare and the curve bulges outside the linear rule — the web is doing nothing useful for the axial force and everything for the moment. At high axial ratios the two compete directly and the small moment costs a disproportionate amount of the axial capacity.

Columns in the lower storeys of a building are at high axial ratios by definition, which is exactly where the nominal moment is most expensive. And because those columns are also the slenderest relative to their load, the moment is amplified.

Amplification does not discriminate between the moments it is given. A moment on a compressed member is multiplied on its way through it by a factor that grows with the ratio of the axial force to the member’s buckling load, and the nominal moment is multiplied along with every other moment in the same load case. So the two effects compound in the same direction and at the same place: the storey with the highest axial ratio has the most expensive interaction and the largest amplifier, and the eccentric moment arrives there having been computed from a lever arm nobody re-measured.

The base, where the eccentricity is an output #

At the bottom of the stack the same question appears with the roles reversed: instead of the connection deciding where the force acts, the force decides where the connection acts.

On a plate carrying 800 kN with 120 kNm on it the resultant sits 150 mm from the centre against a kern of 83 mm, so the plate is in partial contact and the bearing is a triangle rather than a trapezium.

A base plate in full contact has its reaction wherever the pressure resultant is, and that moves as the moment changes. Past the kern the plate lifts on one side, the bearing shortens, and the reaction’s line of action moves further out — so the effective eccentricity grows with the moment rather than staying put. Only when the moment is large enough do the holding-down bolts start working, and on the plate drawn that is not until 200 kNm, well past the 67 at which it first lifts.

Where the structure meets the ground is the essay about that transition. The point here is narrower: the eccentricity at the base of a column is not a detailing dimension, it is a solved quantity, and it is the one place in the stack where the arrow’s position comes out of the arithmetic rather than off a drawing.

The same effect in a truss #

The argument is not specific to building frames. Anywhere a member’s line of action misses a joint’s working point, the same couple appears.

In a welded truss the members’ centrelines are usually made to meet at a point precisely so that no such couple exists, and the residual effect is the joint that is not a pin — continuity rather than eccentricity, worth a few per cent of bending stress. Where the centrelines cannot be made to meet, because a bolted connection has to clear something, the couple is first-order and much larger than the continuity effect.

The clearest place to price that is a bracket, where the eccentricity is undisguised and the bolts have to pay for it directly.

That last fraction is the one to carry away from the whole section. Two thirds of the governing bolt force is bought by a lever arm, not by the load, and the arm is a dimension somebody chose for clearance. The column’s version of the same arithmetic is less visible only because the couple is handed to a member rather than to a fastener — and a member, unlike a bolt, has no line on its check sheet where a couple of this kind belongs.

The larger eccentricity, which nobody applied #

The beam’s reaction is 80 kN and its eccentricity is 200 mm. The column’s own axial load is 1,500 kN, and if it is eccentric by anything at all the arithmetic changes character.

It usually is, at every splice. A column stack reduces in section as it rises, and the two sections have to be aligned somehow. Aligned on their centroids, the load passes straight through and nothing happens; aligned on a face, which is what a flush line of cladding or a plasterboard reveal requires, the centroid moves by half the difference in depth. A 254 UC spliced to a 203 UC and set flush on one flange is offset by

$$\frac{254 - 203}{2} = 25.5\ \text{mm}$$

and 1,500 kN through 25.5 mm is 38 kNm — more than twice what the beam delivered, applied at the same joint, generated by a decision taken for a reason that had nothing to do with structure.

The two figures are the same picture with different numbers in it, and the comparison is the argument. An eccentricity worth worrying about is not a large distance; it is a distance times whatever happens to be passing through it.

The same thing happens in masonry and concrete walls, at a scale that gave the subject its rules of thumb. A wall that steps in thickness at a floor and keeps one face flush has its axial load offset by half the step, which for a 100 mm reduction is 50 mm — and it is where the conventional allowance of an eccentricity of a sixth of the thickness comes from. The convention is not about the floor’s reaction at all; it is about the wall above not sitting on the middle of the wall below.

Two things make this the harder of the two effects to catch. It is proportional to the full axial load, which grows down the building while the beam reactions stay roughly constant per floor — so it is largest exactly where the axial ratio is already worst. And the offset is set at a splice, which is a location chosen for erection convenience and drawn by a fabricator, on a member whose analysis assumed a single straight line from base to roof.

The check is the same sentence as before, asked of a different arrow. For every reaction, ask where it lands — and for every member, ask whether the one above it is sitting on its centre.

Why it is called a nominal moment, and what that hides #

The name given to this quantity in practice — a nominal moment — is doing a lot of work, and it is worth unpicking because it points at two different things.

It is nominal in the sense that the eccentricity is a convention rather than a measurement. Nobody has established that the reaction acts 100 mm from the face; the number is a stand-in for a distribution of bearing pressure over a cleat whose real resultant depends on how much the cleat bends, how tight the bolts are and how much the beam end has rotated.

It is also nominal in a second and more useful sense: the moment is real, but the structure’s ability to shed it is large. A joint that cannot quite carry the moment it is given simply rotates a little more, the beam’s end rotation changes by a fraction of a milliradian, and the couple redistributes. This is a redundant structure and the couple is a compatibility quantity, not an equilibrium one — the stiffest path takes the load, and a path that softens takes less.

That second sense is the reason the whole effect can be treated with a rule of thumb rather than a calculation, and it is also the reason the rule of thumb has to be conservative. A quantity that depends on relative stiffnesses is a quantity that can be anywhere between zero and its largest plausible value, and there is no analysis anybody runs that would tell them which.

The honest summary is that the nominal moment is a bound on something uncertain, applied to make sure a column has a little bending capacity in reserve. What it protects against is not really the eccentricity; it is everything a connection does that a pin does not.

Where the model stops #

The eccentricity is a convention. Nobody measures where the reaction acts on a cleat. The value used — 100 mm from the face, or the bolt line, or half the seat — is a rule chosen to be a safe overestimate, and the real point of application moves as the connection deforms.

The couple may be resisted by something else. A column bolted into a stiff floor plate at each level, or restrained by cladding rails, has more than two branches to share the moment among, and a frame model containing none of them gives the whole couple to the column.

The column is not straight either. Everything above adds a moment to a member that already has one from its own lean, and the two have to be combined with attention to their signs. A column bowed toward the side its beam frames in on is a worse case than the same column bowed the other way, and neither is what an analysis with an eccentricity applied in one direction reports.

Cancellation is assumed rather than checked. The reasoning that an internal column gets nothing depends on the two beams being loaded simultaneously and equally, which is a load case rather than a fact. It is also the reason the effect is invisible on so many projects: it genuinely does cancel most of the time.

What the picture cannot show #

A frame model draws members as lines and connections as points, so there is no dimension anywhere in it corresponding to the quantity this essay is about. The eccentricity lives on a fabrication drawing, produced later, by somebody working from the frame model’s outputs.

That ordering is the practical difficulty. The moment depends on the connection detail; the connection detail is chosen after the analysis; and nothing in the analysis has a place to record the assumption that was made. A change from a fin plate to an end plate, or a column turned through ninety degrees for erection reasons, changes a moment nobody re-computed.

Nor does the picture show the direction. The couple acts about whichever axis the beam frames in on, and a column in a building frame receives beams about both. The load case that matters is frequently the biaxial one — a moment about each axis at the same time, from two different beams, on a section whose minor-axis capacity is a fraction of its major.

The generalisation #

The habit worth carrying is that a line of action is part of a force, and a model made of lines has thrown that part away.

How much of the moment is genuinely thrown away is a question with an exact answer, and it is not “all of it”. Take the moment of a force system about a run of different points and part of it changes and part of it cannot.

So a choice of point is a choice about the perpendicular part only. The parallel part is a property of the system, and no reduction, no node and no idealisation can move it. That is worth knowing precisely because it says what the eccentric column moment is not: it is the perpendicular part, the part that depends on where the arrow was put, and it can be made to vanish at one point and reappear at another — which is exactly how a model made of nodes loses it.

A building frame is drawn flat, one plane at a time, so the wrench never appears on a drawing. It appears in the building: beams frame into a column about both axes, at different levels of the same connection zone, and the system of forces around a real column joint is neither coplanar nor concurrent. The couple this essay is about is one component of it, taken about one axis, in one of the two planes somebody drew.

Every idealisation in this collection does something similar. A support becomes a point and its width comes back as a set of corrections. A connection becomes a node and its size comes back as an eccentricity. A distributed load becomes a resultant and its extent comes back as a difference in the moment diagram. In each case the simplification is exact for the global equilibrium it was made for and leaves a residue that has to be added back by hand.

What makes the eccentric column moment the most persistently missed of the family is that the residue does not look like a residue. It is a moment in a member, of the same kind as any other moment in a member, and it is added to a design that has no line in it where such a thing could be written down. The check is a sentence rather than a calculation: for every reaction, ask where it lands.