The moment that was moved on purpose

Run a beam over two equal spans, put a uniform load on it, and solve. The stiffness method returns a hogging moment of $wL^2/8$ over the middle support and a sagging peak of $9wL^2/128$ in each span — for the beam drawn here, 240 kNm and 135 kNm. Those are the moments the beam has.

They are not the moments the beam must be designed for.

A designer may reduce the support moment by a stated fraction, take whatever span moments statics then requires, and provide sections for those instead. The support moment falls to 168 kNm, the span moment rises to 163, and the largest moment anywhere on the beam has gone from 240 to 168 — thirty per cent less section, for the same beam under the same load. Nothing was approximated and nothing was neglected.

Which free body produced the number #

Cut a single span out of the pair and draw it with whatever moments the rest of the structure applies at its ends.

The outer end is a simple support and carries nothing. The inner end carries the support moment $M_s$, hogging. Everything between them is decided by statics alone: the moment at a distance $x$ is the free moment of a simply supported span plus the straight line joining the two end values,

$$M(x) = \frac{wx(L-x)}{2} - \frac{M_s x}{L}$$

and the peak is wherever the derivative vanishes, at $x^* = L/2 - M_s/wL$.

That is the whole of it. $M_s$ appears as an input, not as an output. The elastic analysis supplies one value for it; any other value produces a perfectly good set of internal forces in equilibrium with the same load, in a beam with the same span and the same supports. What the elastic analysis added — and what redistribution discards — is the requirement that the beam’s two spans have the same slope where they meet.

Why a different diagram is allowed #

The permission comes from the lower-bound theorem, and it is worth stating in the form that makes the licence obvious.

If a set of internal forces can be found that is in equilibrium with the applied load and nowhere exceeds the strength of the material, the structure will not collapse under that load.

It does not say the structure will distribute its forces that way. It says only that it could, and that is enough, because a structure with somewhere to go does not collapse. The same theorem licences a strut-and-tie model in a region where sections do not exist, and a chosen crack angle in a cracked web. Moment redistribution is that argument applied to the tamest structure in the subject.

The condition it attaches is not a small one. The structure has to be able to reach the distribution it is being designed for, which means the sections that are being asked to shed moment have to rotate while holding it.

The mechanism the redistribution is heading toward is the end of that road: a hinge over the support and one in the span turn the beam into a mechanism, and the load at which that happens is its collapse load. Every intermediate distribution between the elastic one and that one is available, and the design is a choice of where to stop along the way.

The redistribution that is the plastic design #

Push the redistribution until the two moments are equal and something exact happens.

The search here is a bisection on the fraction $\beta$, and it stops at 31.4 per cent, where the support and span moments are both $wL^2/11.657$. That denominator is not a coincidence and it was not looked up: it is the collapse coefficient of a propped cantilever, computed elsewhere on this site by searching over the position of the sagging hinge, and it appears here from an entirely different question — where do two moments become equal?

The reason the two questions have one answer is that a beam of uniform capacity collapses when both hinge locations reach that capacity at once, and the balanced design is precisely the one that provides no more than either of them needs.

Its collapse load, computed from the mechanism rather than assumed, comes out at exactly the design load. Every less-redistributed design has a margin: the fully elastic one collapses at 1.46 times its design load, because the section it provides is set by a support moment the beam sheds long before it fails.

What it costs, in radians #

Reducing the support moment by $\Delta M$ is releasing the redundant by that much, and the released structure opens up a relative rotation at that support of

$$\theta = \Delta M \cdot \frac{2L}{3EI}$$

— each span contributing $L/3EI$, the flexibility of a member with a moment at one end and a simple support at the other. For the beam drawn, thirty per cent of 240 kNm over a stiffness of 42,000 kNm² gives 9.1 milliradians.

Nothing about plasticity enters that calculation. It is the compatibility the elastic solution was enforcing, measured. The support section has to turn through that angle while continuing to hold 168 kNm, and whether it can is a question about the section rather than about the beam.

Measured on real connections rather than on an idealised section, that capacity varies enormously: two joints of the same nominal moment can differ several times over in the rotation they will deliver before they shed it, and a joint that reaches its moment and stops is a joint that cannot redistribute anything at all. The demand is computed exactly and the capacity is known roughly, which is the ordinary shape of a ductility check.

Why the limit is a section classification #

Codes cap redistribution at something like thirty per cent, and the cap is usually met as a number in a clause. It is a rotation requirement wearing a percentage.

The rotation a hinge can deliver depends on how much of the section can yield before something local goes wrong, and what goes wrong is the plate buckling before the fibre yields. A stocky section reaches its plastic moment and holds it through a large rotation. A slender one reaches a smaller moment and loses it immediately. So the permitted redistribution is tied to the section class, and the tie is not administrative: it is the same rotation, arrived at from the other end.

Which class a section falls into depends on the grade of steel as well as on its dimensions. A higher yield stress makes every plate relatively more slender, because the class limit is a ratio of the plate’s own dimensions to a length that scales with $\sqrt{235/f_y}$ — so the stronger steel is the one that redistributes less, which is the opposite of what the word stronger suggests.

There is a second reason the cap exists and it is about serviceability rather than collapse. A beam designed for a redistributed diagram is not redistributing under working loads: at any load below first yield the beam has the elastic distribution, whatever the design assumed. So the support region is carrying more moment than it was designed for, and cracks there open wider than the calculation suggests. The redistribution is a promise about behaviour at the ultimate limit state and says nothing about the beam a person is standing on.

The two hundred years it took to be allowed #

The elastic analysis of a continuous beam is older than the permission to ignore it by most of a century, and the order in which the two arrived is worth a paragraph.

Continuous-beam theory was complete by the middle of the nineteenth century, and it arrived as a triumph: a method that could say what a redundant structure carries, where statics alone had been able to say nothing. The moments it produced were treated as facts about the beam, because that is exactly what they had been produced as. A design that used any other set of moments was, straightforwardly, a design that had got the analysis wrong.

What changed was not the analysis but a proof about collapse. The plastic theorems of the nineteen-fifties said that a structure with somewhere to go does not fail at the first yield, and that any equilibrium set of internal forces the material can sustain is a safe design. That turns the elastic diagram from the answer into an answer, and it does so without contradicting a single line of the elastic derivation — the elastic moments remain exactly right for a beam whose every section is still elastic, which the beam is, right up to the load nobody designs for.

The residue of the older view is visible in how redistribution is written down. It is expressed as a reduction from the elastic diagram, capped as a percentage of it, rather than as a diagram in its own right. The elastic solution is still the reference even in the clause that permits departing from it.

The same freedom, without the plasticity #

There is a version of this that needs no yielding at all, and it is worth separating because it is often confused with the one above.

A redundant structure’s internal forces depend on the relative stiffnesses of its members. Make the support region less stiff than the analysis assumed — because it has cracked, because the section is smaller there, because the joint is not rigid — and the moment moves away from it elastically, with nothing anywhere past yield. That is not redistribution in the technical sense; it is a different analysis of a different structure. But it produces the same picture, and it means the elastic diagram is sensitive to a stiffness nobody measured.

A joint made of springs in series is where the assumed rigidity actually comes from, and none of those springs is infinitely stiff. A connection with a real stiffness makes the frame’s moments a function of that stiffness, and the elastic distribution moves without anything having yielded.

The practical consequence is a reason to be sceptical of the elastic diagram in both directions. It is not a measurement. It is the output of an analysis whose inputs include a set of stiffnesses that are estimates, and a beam whose support region has cracked has already redistributed some of its moment without being asked.

The stiffness that has already changed is the commonest instance of it. A cracked section is perhaps a third of the uncracked one’s stiffness, and the hogging region over a support cracks first because that is where the moment is largest — so the real structure begins shedding support moment at a fraction of the design load, with no plasticity anywhere.

The frame, where the shed moment has to land somewhere #

A beam is the easy case, because the moment taken off the support goes into the same member. In a frame it does not.

A beam end that sheds 30 per cent of its hogging moment is a beam end whose joint is no longer in balance with what it was: the column above and below were sized for a share of the elastic beam moment, and the moment they receive falls with it. That is a saving, and it is also a hazard, because the beam is now being designed on the assumption that it will hinge at the joint — and if the columns are the weaker members, the hinge forms in them instead. A mechanism with hinges in the columns of a storey is not a redistribution; it is a storey collapse.

This is the whole of what capacity design is about, and it turns up here in miniature: choosing to redistribute is choosing where the structure will yield, and a choice about where a structure yields is only worth anything if the rest of it is strong enough to make the choice come true.

The frame the same argument moves into has no such tidy accounting. Redistributing a beam’s support moment changes what the joint delivers to the columns, and the columns were sized from the diagram that has just been changed. In a beam the shed moment goes into the span; in a frame it goes into whichever member is least able to refuse it.

The shear does not get the same licence #

Moving the support moment moves the reactions, and moving the reactions moves the shear. That much is arithmetic: with the outer end free, the end reaction is $wL/2 - M_s/L$, so reducing $M_s$ from 240 to 168 kNm raises it from $0.375wL$ to $0.4125wL$ and drops the middle support’s reaction from $1.25wL$ to $1.175wL$. The shear at the inner face falls by 6%.

It is tempting to bank that, and it should not be banked. The theorem that licensed the moment redistribution does not extend to the shear, because its condition does not hold there.

The lower-bound argument requires the structure to be able to reach the assumed distribution, and reaching it means a flexural hinge rotating at constant moment. Shear failure in a web is not like that: it arrives with little warning, sheds capacity as soon as it starts, and supplies none of the rotation the argument is trading on. A beam that redistributes its moments does so by yielding in bending; the same beam’s shear capacity has no comparable mechanism, so there is no basis for assuming the shear will politely rearrange itself to match.

The practical rule follows, and it looks inconsistent until the reason is stated: design the sections for the redistributed moments and check the shear on the elastic values — or on whichever of the two is larger at each station. The same beam is then being designed against two diagrams that do not correspond to one state of the structure, which is uncomfortable and correct. It is the same reasoning as an envelope assembled from load cases that never coexist: each check is made against the worst credible version of the quantity it is about, and the collection of them describes no single instant.

Redistribution runs both ways #

Everything above reduces the support moment, which is what a designer usually wants because the support moment is the larger one. The theorem does not care about the direction.

Raise the support moment by 30% instead — to 312 kNm — and statics drops the span peak to 109. The beam is still in equilibrium with the same load, still designable, and the hinge that has to rotate is now the one in the span rather than the one over the support.

It is a worse deal on this beam, because the peak moment anywhere has gone from 240 to 312 and the section is set by the peak. It is the right deal whenever the span section is the constrained one and the support section is not: a beam whose midspan depth is fixed by a service running under it, a precast unit whose section is what the mould makes, a composite member whose sagging capacity is limited by the slab and whose hogging capacity is not. In each of those the span moment is expensive and the support moment is cheap, and moving moment toward the support is the economy.

The rotation demand moves with it, and it moves to the harder place. A hinge over an interior support is in a region of steep moment gradient, so its plastic length is short and the curvature has to be concentrated; a hinge in a span sits in a region where the moment is nearly stationary, so the yielded length is long and the same rotation is delivered at lower curvature. Which means upward redistribution is, if anything, the more comfortable of the two to justify — and it is the one nobody thinks of, because the elastic diagram is always presented as a thing to be reduced.

Where the model stops #

The two spans are identical, and the arithmetic of the balanced case depends on it. A pair of unequal spans has a support moment that is not shared symmetrically, the balanced redistribution is different on each side, and there is no single $\beta$ that equalises everything.

The rotation is calculated on the released elastic structure. The real hinge is a finite length of beam whose curvature is distributed over it rather than concentrated at a point, and the rotation it delivers depends on that length — which depends in turn on the shape of the moment diagram near the support, on how much of the length is cracked, and on the bar spacing. The 9.1 milliradians is a demand computed exactly and compared with a capacity known roughly.

And the load is one load. A beam whose imposed load can sit on either span has several diagrams rather than one, and redistribution has to be applied to each of them separately and consistently — which is more constraining than it sounds, because the diagram that governs the support and the one that governs the span are different load cases.

What the picture cannot show #

Both curves are drawn as though they existed at once. They do not. The elastic one is the beam’s state at low load; the redistributed one is a state it reaches only near collapse, if the sections deliver the rotation. In between there is a beam that is partly one and partly the other, and the figure has no way to draw a sequence.

Nor can it show what is happening in the support section while the moment is being shed. The picture is of bending moments, and the event is a rotation at constant moment. The one quantity the whole argument turns on is the one quantity a bending-moment diagram does not contain.

The assumption the figure rests on #

The flexural rigidity is constant along the beam, and the rotation quoted depends on it directly. In a reinforced concrete beam it is not constant: the hogging region cracks and the sagging region may not, so the real structure’s stiffness distribution favours redistribution before anybody chooses any. Taking $EI$ as uniform makes the computed rotation an overestimate of what has to be delivered plastically and an underestimate of what has already happened elastically, and the two errors do not cancel.

None of that touches the bound itself. The diagram is an integral of the load, so the moment curve’s curvature is the load and no redistribution can change it: the parabola between the end values is fixed, moving one end moves the whole curve rigidly, and the free moment is invariant. That single fact is what makes the identity on the first figure hold at every $\beta$.

The ladder from here #

Later rungs on this anchor: redistribution in a frame, where the moment shed from a beam end arrives in a column that was not designed for it. The interaction with pattern loading, where each arrangement has to be redistributed separately and the envelope of redistributed diagrams is not the redistribution of the envelope. Redistribution in a prestressed member, where the secondary moments are themselves a self-equilibrating field and the question of what may be redistributed has an extra term. The rotation capacity of a real hinge, measured rather than classified. And the limit-state version of the same argument, in which the elastic analysis is abandoned entirely and the design is a mechanism from the start.