Two of these move and the third cannot

There is one quantity in a continuous beam that no analysis can change, and it is not a small one. It is the whole bending moment the span has to carry.

Cut a single span out of the beam — anywhere, out of any beam, of any number of spans, with any stiffnesses in it. Draw it as a free body: the load along it, a shear and a moment at each end. Take moments about one end and the other end’s shear falls out; substitute it back and the moment at mid-span comes out as

$$M_{mid} = \frac{wL^2}{8} - \frac{M_{left} + M_{right}}{2}$$

which rearranges to the statement this essay is about. The mid-span moment plus the average of the two end moments is the free bending moment of the span, and the right-hand side contains the load and the span and nothing whatever else.

Which free body produced the number #

One span, cut at both ends, with the internal actions the cuts reveal drawn on it — and nothing else. That is the whole derivation, and it is worth being explicit about what is not in it.

There is no stiffness in the free body, because stiffness is not a force. There is no compatibility, because a free body does not have to fit anything. There is no information about the spans on either side, because they were cut away and replaced by the actions they apply. And there is no assumption about how the beam behaves — elastic, cracked, yielding, creeping — because the only statement made is that the forces on a body at rest add to nothing.

So the identity holds for any beam that is standing up. It holds for a reinforced concrete beam whose supports have cracked and whose stiffness has fallen by a factor of four; it holds for a steel beam with a plastic hinge over each support; it holds for a beam whose middle support has settled twenty-five millimetres. The free body is a choice, and this is the choice that isolates the one thing stiffness cannot touch.

The picture: a parabola with a line under it #

The clearest way to hold the identity is geometric, and it is the way it was taught before it was a formula.

Draw the free bending moment diagram for the span — a parabola of height wL²/8. Now draw a straight line joining the two end moments, hanging below the axis. The continuous beam’s moment diagram is the parabola measured from that line. Raise the line and the mid-span ordinate falls by exactly the amount the ends gained; tilt the line and the diagram tilts with it. The parabola never changes, because the parabola is the load.

That is why the identity is written with an average of the two end moments: the line’s height at mid-span is the average of its two ends, whatever its slope. And it is why a beam with unequal end moments has its maximum sagging moment somewhere other than mid-span — the identity is about the ordinate at the middle, not about the peak, and the two coincide only when the line is level.

What continuity is actually worth #

Continuity has a real value and it is not the one usually claimed for it.

A three-span beam at 7 m and 5 kN/m has a free moment of 30.6 kNm in every span. Continuous, it carries about 25 kNm over the interior supports and 20 in the end spans — the total is unchanged, and the largest single number has fallen from 30.6 to 25. That is a 19% reduction in the peak, which is worth having, and it comes from splitting one number into two rather than from making anything smaller.

Where continuity really pays is deflection, and that is a different quantity with a different arithmetic: the end span of a continuous beam deflects about 40% of what the same span deflects simply supported, because the end moment curves it back. Span to the fourth is why that is worth so much more than the strength gain, and stiffness is not strength is why it is usually the deflection that was governing anyway.

And there is a cost on the other side of the ledger. A continuous beam has a hogging moment over its supports, which puts the tension in the top of the section — where, in a slab, it is exactly where the traffic is, and where in a concrete beam it is the face with the least cover. The moment over the support is that trade in full.

Why redistribution is legitimate, and how much of it there is #

Here is the identity doing the work it is most often used for.

Take the elastic solution and decide to design the supports for less than it says — thirty per cent less, say, on the grounds that the section there will yield and shed moment before anything else fails. What must happen at mid-span? The identity answers it exactly: the mid-span moment must rise by the average of what was taken off the two ends, and by precisely that. There is no room for judgement in the number and no code table involved.

That is the whole basis of moment redistribution, and it is a lower-bound argument. Two ways of being wrong sets out the theorem: a set of internal forces in equilibrium with the load, nowhere exceeding capacity, is safe. The redistributed diagram is in equilibrium — the identity certifies it — so a beam proportioned for it will carry the load, provided the support section can rotate enough to get there.

So the number a code prints — twenty per cent, thirty per cent, forty — is not an allowance about moment at all. It is a statement about ductility: how far a section can rotate before it stops carrying its plastic moment. The moment that was moved on purpose is the rotation side of the same operation, and what is left after the first fibre yields is what decides whether the section has it.

A support that moves, and a beam that does not notice #

The severest test of an identity of this kind is a moment field that arrives with no load attached to it, and a settled support is exactly that.

Drop the middle support of a three-span beam by twenty-five millimetres and the beam picks up a complete set of bending moments — hogging over the settled support, sagging on either side, proportional to EI and present with no applied load whatever. The support that moved is the case in full, and it is the clearest demonstration there is that a redundant structure knows something statics does not.

Add that field to the load field and check the identity again. It holds, to a part in a billion. The settlement moment is itself in equilibrium with nothing, so it contributes zero to the free moment of the span — which means it must change the end moments and the mid-span moment by amounts that cancel in the sum. Both terms move; the total does not.

Pattern loading, and the envelope the identity does not cover #

A continuous beam is not designed for one load case. It is designed for an envelope — the worst moment at every section over every arrangement of imposed load the building might see — and the identity has something exact to say about that too, and something it cannot say.

What it says exactly: each arrangement satisfies the identity separately. Load every span and the support moments are large and the span moments small; load alternate spans and the loaded spans’ mid-span moments rise while the support moments fall. Both are equilibrium sets, both obey the identity, and the second one’s larger span moment is bought with a smaller support moment in precisely the amount the arithmetic requires.

What it cannot say: the envelope is not one of those sets. It is the outer boundary of all of them, assembled by taking the worst ordinate at each section from whichever arrangement produced it — and that curve is in equilibrium with no load case at all. The envelope is not a structure is the whole of that warning. So the sum of the envelope’s mid-span ordinate and its envelope support moments is greater than the free moment, sometimes by a fifth, and a reader who checks the identity against an envelope and finds it broken has found the envelope rather than an error.

That is not a defect of the envelope — it is what an envelope is for. But it is the reason a design check made against an envelope cannot be closed by a statics argument, and the reason the identity is a tool for reasoning about a case rather than about a design.

The same identity, on a flat slab #

The place the identity is used most and named least is the design of a flat slab, where it is the total static moment.

A panel of a flat slab spanning between columns is a beam of width equal to the panel’s other dimension, and the total moment it must carry between two supports is the same free moment as any other beam: the load, times the width, times the clear span squared, over eight. Nothing about the slab’s stiffness, its column stiffnesses, its drop panels or its reinforcement layout can alter that number.

What everything else decides is the split — how much of the total goes over the columns and how much goes across the middle of the panel, and how each of those divides again across the panel’s width into a column strip and a middle strip. The direct design method is a table of those percentages, and it is legitimate for exactly the reason redistribution is: the percentages add to one hundred, so whatever they are, the answer is in equilibrium.

That is a much stronger statement than it first appears. It means a designer who distributes the moment differently from the table — for a slab with an opening, an irregular column grid, a heavy line load — is not outside the rules. They are outside the table, which is a different thing, and the identity is what tells them so.

Why it was ever in doubt #

It is worth asking why a statement this simple needed a name at all, and the answer is that it was arrived at from the wrong end for about a century.

The hand methods of the nineteenth and early twentieth centuries computed continuous beams by solving for the redundants — the three-moment equation, the force method, moment distribution — and every one of them produces the support moments first and the span moments afterwards, by subtraction. From inside that arithmetic the identity is not a theorem; it is a step, and one that looks like a consequence of the analysis rather than a constraint on it. It is very easy to spend a career computing the sum without ever noticing that it never changes.

The reverse view arrived with plastic design in the 1950s, when the question stopped being what are the moments and became what set of moments will do. Asked that way the identity is the first thing anyone writes down, because it is the only constraint there is: choose any pair of support moments and the span moment is decided, and the design problem is a choice on a line rather than a calculation with an answer. Choose what to take away is the same reorientation applied to the force method, and it is the more useful way round for anybody doing the choosing.

Where the model stops #

The load was uniform. For a point load the free moment is Pab/L rather than wL²/8 and the identity’s right-hand side changes with it, but nothing else does — the derivation never used the shape of the load, only its statical moment about the ends. A span carrying a load nobody has drawn has an identity nobody can write down, which is an argument for drawing it.

The span was straight and prismatic. A haunched beam has the same total, and the ordinate at mid-span is still the parabola less the line — but the critical section has moved, because the depth varies along the span and the worst utilisation is no longer where the worst moment is. The section that changes along the span is that shift.

The supports were points. Over a wide column the hogging moment falls between the column faces, and the value used in design is taken at the face rather than the centreline — which reduces the support term and, by the identity, cannot reduce the total. The support that is not a point is the correction and the reason it is legitimate.

And the beam was in one piece. The identity is about a span between two cuts. Put a hinge inside the span and there are two free bodies, each with its own identity, and the arithmetic is different — which is what an articulated deck is doing, and why where the structure is allowed to move changes the moments as well as the movements.

The generalisation #

What is worth carrying away is a habit of asking, of any structural quantity, whether it is a statics quantity or a stiffness quantity — because the two behave completely differently under everything a real structure does.

A statics quantity is fixed by the load and the geometry. It does not care about cracking, creep, settlement, temperature, sequence or the analyst’s model, and no amount of stiffening or softening anything will move it. A stiffness quantity is fixed by the relative stiffnesses of the parts, moves with every one of those effects, and is only ever known as well as the model is.

The total moment in a span is the first kind. The split between support and mid-span is the second. A designer who spends effort on the second and takes the first for granted has it exactly right; one who does the reverse is refining a number that was never uncertain while trusting one that was.

The same test sorts most of the subject. The thrust in a two-pinned arch is a stiffness quantity; the total moment its line has to accommodate is a statics one. The force in one bolt of a group is stiffness; the sum of them is statics. The torque a spandrel beam attracts is stiffness — the torsion that goes away if it is let — while the torque on a cantilever canopy is statics and will not go anywhere at all. Knowing which is which is most of knowing what an analysis is for.