The moment that was shed has to land

Moving a moment on purpose is one of the few genuinely free things in structural design. Take a fraction off the support moment of a continuous beam, let statics supply what follows, and the section needed falls in exact proportion — paid for in rotation rather than in material.

That is the argument on a beam. This page is about what happens to it when the support is not a support.

The saving is linear and the demand is not #

Before leaving the beam, one property is worth having, because it is what makes the technique attractive in the first place.

The free moment $wL^2/8$ is a property of one span as a free body and is untouched by anything. Whatever the end moments are, the sagging ordinate at mid-span is the free moment minus their average, so a reduction $\beta$ at the support raises the span moment by an exactly compensating amount. Both the saving and the rotation are therefore linear in $\beta$.

What is not linear is whether the section can supply the rotation. A section’s ability to turn while holding its moment is what a classification is about, and it is a step function: a section either has the class or does not. So the benefit is continuous and the permission is discrete, which is why codes cap $\beta$ at a value that depends on the class rather than scaling it.

Where the shed moment goes on a beam #

On a continuous beam, the answer is comfortable: it goes into the adjacent spans, and there is room for it.

Shed 30 per cent of the 243.0 kNm hogging and the span moments rise from 194.4 toward the free value, the reactions shift slightly, and every span still satisfies its own free-moment identity. The structure has moved from one equilibrium state to another and nothing outside the beam has noticed, because the supports are supports: they take whatever vertical force they are given and have no opinion about moment.

That is the case every textbook presents, and it is the only case in which redistribution is a purely local decision.

The joint is where it stops being free #

Replace the middle support with a column and the arithmetic changes.

At a beam-to-column joint the moments must balance: the beam’s end moment is delivered to the column, and if two beams arrive the difference between them is. Reducing the beam’s end moment by 91 kNm does not delete 91 kNm. It transfers it, and the only place it can go is the column, whose section was chosen from the elastic analysis that has just been set aside.

There are three defensible responses and they are not equivalent.

Re-solve the frame with a hinge at the beam end. This is the honest version: insert a release, apply the load again, and read off what the column now carries. The answer is usually a larger column moment and a different distribution down the storey.

Redistribute the beam and leave the column at its elastic value. This is what is usually done. It is safe only if the column’s elastic moment happens to exceed what the redistributed state delivers, which is not guaranteed and is rarely checked.

Forbid redistribution at joints with columns. Some codes do exactly this for the beams framing into edge columns, where the column is small and the transfer is proportionally large.

The general statement is uncomfortable and worth making plainly: redistribution is a change to the structure’s internal force state, and an internal force state is global. Adjusting one ordinate of one diagram and leaving the rest is not a redistributed solution; it is two solutions to different problems, printed on one drawing.

Which free body produced the number #

Two free bodies, and the second is the one that is usually skipped.

The span. Cut one span out with a vertical section at each end. Crossing the cuts are two moments, two shears and the load in between. Taking moments about mid-span gives the free-moment identity, which contains no stiffness at all — and it is what guarantees that any pair of end moments has a legitimate span moment to go with it. That is the whole licence for redistribution and it is the lower-bound theorem in its smallest possible form.

The joint. Cut a small block containing the beam-column intersection. Crossing its faces are the beam’s end moment, the column’s moments above and below, and the shears and axial forces that go with them. Rotational equilibrium of that block says the moments sum to zero, and it says so whatever analysis produced them.

The second free body is what makes redistribution non-local. The first says a beam may take any pair of end moments; the second says whatever it takes is delivered to something. Presenting redistribution using only the first is why the technique is remembered as free.

The envelope cannot be redistributed #

There is a second and more common mistake, and it is about which diagram the reduction is applied to.

An envelope is not a state of the structure. It is the outer contour of several states, and no arrangement of load produces it. Redistribution is a move from one equilibrium state to another, so it can only be applied to something that is a state.

Applying $\beta$ to the envelope is therefore not conservative or unconservative in any predictable direction — it is a move from a curve that is not an equilibrium field to another curve that is not one, and the free-moment identity that licensed the whole technique has been abandoned at the first step.

The correct procedure is unglamorous: redistribute each arrangement separately, and build a new envelope from the redistributed diagrams. The result differs from the redistributed envelope, and it differs most where the arrangements disagree most — which is exactly at the interior supports where the redistribution is being applied.

What has to move, and how far #

The rotation is the price and it is worth locating precisely.

That figure says why redistribution works and also what it is. Sweeping the stiffness of a member moves the moments along the same one-dimensional family that redistribution moves them along. Redistributing by 30 per cent is equivalent to having analysed a structure whose member stiffnesses were different, and the hinge rotation is the difference between the two structures’ compatibility.

That equivalence is the most useful way to think about the demand. A section that yields is a section whose stiffness has fallen, so a redistributed structure is the elastic structure with one member softened. The rotation is what the softening produces, and the section has to be able to deliver it while still holding its moment — which is why the requirement is on rotation capacity and not on strength.

Why the technique is worth the trouble at all #

Given the bookkeeping, it is fair to ask what redistribution buys, and the answer is not mainly a smaller section.

It moves steel out of a congested place. The support of a continuous concrete beam is where the top steel from two spans, the column bars and the links all meet. Taking 30 per cent off the hogging moment removes bars from the one place in a frame where they cannot be fitted, and adds them to mid-span where there is room.

It equalises the two design moments. The elastic diagram on the beam above has 243 kNm hogging and 194 sagging; at β = 0.3 they become 170 and 218. A member designed for the larger of the two is designed for 218 rather than 243, and a member with the same section throughout is being used efficiently in both regions rather than in one.

And it acknowledges what has already happened. The elastic diagram belongs to an uncracked structure. Redistribution is the designer choosing a state closer to the one the structure will actually be in, rather than detailing for a state it left at a fraction of the design load.

None of those is a saving in material of any size. The saving is a fraction of a percentage of the frame’s weight, and the reason the technique persists is that it makes a difficult detail buildable — which is the same argument a variable truss angle makes about links, reached from bending instead of from shear.

The other way it happens, without being asked #

Redistribution is also something structures do on their own, and the distinction between the deliberate and the accidental is thinner than it looks.

After the first yield a structure redistributes whether or not anybody planned it. Cracking does the same in concrete, at loads far below yield: a cracked region loses stiffness, sheds moment to its neighbours, and arrives at a state the uncracked analysis did not predict. A support that settles does it too, and so does creep.

So an elastic analysis of a concrete frame is a calculation about a structure that does not exist — the real one has cracked where the moments were largest, which is precisely where the analysis said to put the steel. The redistribution allowance is partly a permission and partly an acknowledgement, and codes that cap $\beta$ at 30 per cent are capping the deliberate part of something that has already happened by an unknown amount.

The hand method that was this argument #

Moment distribution is worth putting beside this because it is the same operation done for a different reason. Each cycle releases a joint, lets it rotate, and passes the out-of-balance to its neighbours — which is exactly the transfer described above, applied in small amounts and iterated until the joint is in equilibrium.

The method makes the non-locality unavoidable and visible. Nobody using it could believe that reducing a moment at one joint left the rest of the frame alone, because the next line of the calculation is the carry-over. The matrix hid that, and the modern habit of adjusting one number on a printed diagram is a habit the hand method made impossible.

What a redistributed frame actually needs #

Putting the pieces together, the procedure that is defensible is longer than the one that is usual, and it is worth setting out because the difference is where the risk lives.

One. Analyse the frame elastically, for every load arrangement.

Two. For each arrangement, choose the beam ends to be redistributed and apply β to those ends only.

Three. For each arrangement, restore equilibrium at every affected joint — which means giving the columns the moment the beams have shed, in proportion to their stiffnesses, and carrying the change on down the storey.

Four. Build the envelope from the redistributed arrangements, and design every member from it.

Steps three and four are the ones omitted. Step three is omitted because the beam calculation and the column calculation are usually done by different people or at different times; step four because the envelope is what appears on the output and is the natural thing to reach for.

The cost of doing it properly is small — it is one more pass of the same analysis — and the cost of not doing it is that a fraction of the frame’s moment has been deleted rather than moved. Redistribution is bookkeeping about where a moment is, and bookkeeping that loses a quantity is not conservative in any direction.

How large the transfer actually is #

It is worth putting a number to the column’s share, because it is not obvious whether the effect is a detail or a governing case.

Take an interior column in a frame with beams either side. Under a symmetric load the two beam end moments are equal and opposite, they cancel at the joint, and the column carries nothing — so redistributing both by the same β changes the column’s moment by nothing at all. The interior column under symmetric load is genuinely unaffected, and that case is common enough to explain why the omission survives.

Now take an edge column, with one beam. The whole of the beam’s end moment is delivered to it, split between the column above and below in proportion to their stiffnesses. Shedding 30 per cent of a 243 kNm beam moment moves 73 kNm off the beam and onto a column pair — perhaps 40 kNm to the upper column, on a member whose elastic design moment may have been 80. A fifty per cent increase, at the joint the code is most likely to permit redistribution at.

Between the two lies the interior column under pattern loading, where the two beams carry different moments and the difference is what the column takes. Redistributing the more heavily loaded beam increases that difference, so the transfer is real and is largest for exactly the arrangement that governs the column.

Where the model stops #

The rotation capacity is asserted rather than computed. Codes tie the permitted $\beta$ to a section class, which is a proxy for rotation capacity based on plate slenderness. The actual rotation a hinge can deliver depends on the moment gradient, the axial force, the confinement and the reinforcement, and none of those is in a class.

The redistributed state is assumed reachable. The lower-bound theorem promises safety if the structure can reach the assumed state, and reaching it means yielding at the support and staying there while the span catches up. A section that loses moment as it rotates has not redistributed; it has failed.

Nothing here computes crack widths. A redistributed member is one whose support region is deliberately worked harder in rotation, and the crack widths there at service load are wider than the elastic design would have given.

Axial force is ignored. A column carrying the shed moment is carrying it with a large axial load, and its rotation capacity at that axial load is much smaller than a beam’s. The member being asked to absorb the transfer is the one least able to hinge.

The transfer is assumed to be resisted elastically. A column given a moment it was not designed for may itself yield, redistributing again — and a chain of redistributions through a frame is a mechanism forming, which is a different calculation with a different theorem behind it.

And the analysis is first-order. Shedding moment from a beam end softens the frame, which increases its sway, which increases the second-order moments in the columns — the transfer described here and the amplification arrive together.

What to carry away #

A moment is never removed, only moved. The free-moment identity licenses moving it within a span; joint equilibrium says where it goes when it leaves one. Any presentation of redistribution that mentions the first and not the second is describing half of an operation.

The thing being redistributed has to be a state. An envelope is not one. Redistribute each arrangement and rebuild the envelope; the two orders of operation give different answers, and only one of them is equilibrium.

And the demand is a rotation, at a member carrying axial load. The section class that permits β was calibrated on beams. A column being asked to absorb the transfer is at the other end of the ductility range, and the permission granted to one is being spent by the other.

The rotation that pays for the redistribution is the same one a compatibility torque needs before it can be shed, and the two arguments have the same structure: a permission granted at the ultimate limit state, priced in a deformation nobody computed.

The ladder from here #

Later rungs on this anchor: redistribution in a prestressed member, where the secondary moments are themselves a self-equilibrating field and there is an extra term in what may be moved. The rotation capacity of a real hinge, measured rather than classified, against the demand a given β generates. Redistribution and pattern loading resolved together, arrangement by arrangement. Redistribution downward — increasing a support moment, which is permitted, unusual and occasionally the right answer. The limit-state version, in which the elastic analysis is abandoned from the start and the design is a mechanism. And the same argument in a slab, where the redistribution is two-dimensional and the yield lines rather than the hinges are what has to be able to turn.

The permission is older than the theory that justifies it. Codes allowed a 15 per cent reduction in support moments from the 1930s, on the empirical grounds that continuous beams did not fail where the elastic analysis said they should, and the plastic theorems that explain why arrived twenty years later. The number has barely moved since, which for a rule with a proof behind it is either reassuring or a sign that nobody has revisited it.