Structural Engineering & Statics
where the load goes
Internal forces

The moment the beam left behind

A beam reaction is drawn arriving on a column's centreline. It arrives on a cleat a hundred millimetres out from the face, and the difference is a couple that goes into the column and has to be shared between the lengths above and below it. Nothing about it appears in a frame model whose members meet at nodes.

Assumes Moving a force, and what it costs, The connection is not a point, and every diagram on this site says it is and The stiffest path takes the load.

17 min read 10 figures The load must go somewhereThe statics of things that do not moveOne support too many

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.

The moment is the force times the distanceOne force applied at five distances from a pivot, with the moment it produces drawn as a bar. The force never changes; only the arm does, and the moment follows it exactly.pivotthe same force of 60, moved along the levermoment about the pivot0.160 × 0.1 = 60.260 × 0.2 = 120.360 × 0.3 = 18
Fig. 1 One force at several distances from a pivot. Nothing about the force changes and the moment follows the arm exactly, which is the whole of what an eccentric reaction does.

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 RR, applied at a point a distance ee 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 RR on its axis plus a couple ReRe, and the two systems are equivalent for every equilibrium equation anybody can write about the column.

A force may be moved anywhere, at the price of a coupleA 80 kN force applied 250 mm off the centreline of a body, and the same force applied ON the centreline together with a couple of 20 kNm. The two systems are equivalent: they have the same resultant force and the same moment about every point in space, so no equilibrium equation written about the body can tell them apart. What they are not is the same loading — the stresses inside the body differ, and they differ over a distance of about the body's own depth. The offset is drawn to a scale that keeps the arrow on the body; the number beside it is the real one.80 kNe = 250 mmas applied: one force, off the line80 kNcouple 20 kNmas replaced: the same force, on the line, plus a couple
Fig. 2 A force applied off the centreline, and the same force on the centreline together with a couple. Every equilibrium equation about the body gives the same answer for both, and the stresses inside it do not.

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=201203/2 + 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.

A beam, its loads and its reactionsA free body: the beam cut away from everything it touches, with the forces that were doing the touching drawn on it. The reactions are computed from the loads, so their arrows are to scale relative to each other.1286.014.0ΣM about one support gives the other reaction; ΣF then gives the first
Fig. 3 Where the reaction came from. Nothing in this drawing says where on the support the arrow lands, which is exactly the information the column needs and 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 ReRe 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.

What each member takes is decided before anything is distributedDistribution factors at every joint of a two-span beam of 6, 6 m. At each joint the out-of-balance moment is shared between the members meeting there in proportion to 4EI/L, so a short span takes more of it than a long one — the factors are properties of the geometry and are written down once, before any arithmetic happens. The fixed-end moments the method starts from are wL²/12: 30.0, 30.0 kNm.joint 01.000joint 10.5000.500joint 21.0006 mFEM 30.0 kNm6 mFEM 30.0 kNmthe factors at a joint sum to one — nothing is created or lost in a distribution
Fig. 4 Distribution factors at a joint. An out-of-balance moment is shared between the members meeting there in proportion to 4EI/L, and the factors are properties of the geometry written down before any arithmetic happens.

The answer is the ordinary one for any out-of-balance moment at a joint: it is shared in proportion to 4EI/L4EI/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.

Moment against rotation, for three real jointsThree connections on one plot, with the classification boundaries for a beam of EI/L = 20000 drawn as rays through the origin. web cleats is pinned, flush end plate is semi-rigid, extended end plate is semi-rigid. The boundaries are multiples of EI/L, so the same joint is rigid on a short stiff beam and semi-rigid on a long slender one.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
Fig. 5 Three real connections, with the classification boundaries drawn as rays through the origin. What counts as a pin is a statement about the beam it serves rather than about the connection.

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.

Two ways to fail, and the curve between themThe exact plastic interaction between axial force and moment for one section of identical area, both normalised by their own squash load and their own plastic moment. The I-section stands 6.0% of its plastic moment outside the straight line at an axial ratio of 0.12. Every section here is symmetric about its centroid, so the equal-area axis and the centroid coincide and it makes no difference which the moments are taken about. The straight line is the rule that says the two capacities share out in proportion, and everything between it and a curve is capacity that rule gives away.00.20.40.60.8100.20.40.60.81moment ÷ plastic momentaxial force ÷ squash loadI-section: 6.0% of Mp outside the linethe straight-line rule
Fig. 6 The exact plastic interaction between axial force and moment for one section, both normalised by their own capacities. The section stands 6% outside the straight line at low axial ratios.

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.

The load that makes itself worseThe amplification of a deflection against the ratio of applied load to buckling load. A structure at half its buckling load deflects twice as far as first-order analysis predicts, and the curve runs away well before the load is reached.00.20.40.60.80246810applied load ÷ buckling load1.2×1.4×first-order analysis says the answer is always 1×one over one minus the ratio
Fig. 7 The amplification a moment gets on its way through a compressed member. It is applied to the eccentric moment along with everything else, and it is largest where the axial force is largest.

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.

A base plate, and when the bolts start workingA 500 × 400 mm plate carrying 800 kN and 120 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 13.33 N/mm², with the holding-down bolts carrying 0 kN. The plate lifts at 66.67 kN·m and crushes at 146.67 kN·m, and the bolts are not needed until 200 kN·m.800 kN120 kN·mresultant at e = 150middle thirdbolt carries nothingPartial contactbearing over 300 mm at 13.33 N/mm² · 66.67% of 20
Fig. 8 A base plate carrying axial force and moment. 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.

The joints are not pins, and this is what that costsA 6-panel Pratt truss solved twice on the same stiffness matrix: once with a moment release at every member end, which is the pin-jointed idealisation, and once with the joints continuous, which is what welding them produces. The axial forces are the same to within a per cent; the bending the second solution adds is worst in member 13, where the bending stress reaches 8.6% of the axial stress. Members are shaded by that ratio.worst secondary bending: 8.6% of the axial stress, in the member markedaxial force there 15.0 kN · end moment 0.01 kNm · slenderness of the member 80the same members, the same loads, the same solver — only the releases differ
Fig. 9 A truss solved twice on the same matrix — once with a moment release at every member end, once with the joints continuous. The axial forces agree within a per cent and the bending the second adds reaches 8.6% of the axial stress.

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 free body of a corner closes to the last digitThe knee of the frame, cut clear of both members. The beam applies 70.0 kN downward, 20.6 kN horizontally and a couple of 49.4 kNm; the column applies 70.0 kN upward, 20.6 kN horizontally and a couple of 49.4 kNm the other way. The beam's end shear of 70.0 kN leaves as the column's axial force of 70.0 kN — vertical equilibrium of this block and nothing else — while the column's shear of 20.6 kN leaves along the beam as axial force. The three residuals are 4.5e-9, 0.0e+0 and 1.4e-14, the last being 2.9e-16 of the corner moment itself: continuity of moment at a rigid joint is not an approximation but an equation the solve satisfied.beamcolumn70.0 kN of shear20.6 kN axial49.4 kNm70.0 kN axial20.6 kN of shear49.4 kNmΣFx = 4.5e-9 · ΣFy = 0.0e+0 · ΣM = 1.4e-14 kNm
Fig. 10 A rigid corner cut clear of both members, with every force on it. The residuals close to fourteen decimal places, which is what makes a joint an equation rather than an approximation.

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.

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.

Where this goes next

the central axispitch 0.480the couple that stays|R| = 67.08|M| on the axis = 32.20their ratio is the pitchthe ring stands for thecouple, and a couple hasno size and no placexyzP₁P₂ Moving a force, and what it costs Every free body on this site begins by putting a force somewhere convenient. That move is free along the force's own line, costs a couple across it, and in three dimensions leaves behind something no choice of point can remove. 100 kNe = 150the dashed ring is the group's centroidworst bolt 50.37 kNSix bolts · direct shear 16.67 kN eachelastic vector method The connection is not a point, and every diagram on this site says it is Every free body drawn here has joined its members at points. Real structures fail at the joints far more often than in the members, and the reason is that a joint is exactly the region the theory behind every other page explicitly excludes. 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.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

Axial forceBase plateBending momentCompatibilityConnectionEccentricityForce coupleFree bodyInteractionJoint classificationLoad pathMoment distributionNominal momentSecond orderStiffness