The redistribution nobody chose

A beam is designed as simply supported. The analysis gives it $wL^2/8$ at mid-span and nothing at the ends, a section is chosen for that, and a connection is detailed — web cleats, a fin plate, something described on the drawing as a simple connection.

The beam then goes and behaves differently, because the connection is not a pin.

The exact result, which is one line #

The redistribution is not hard to compute. A uniformly loaded beam with equal rotational springs at both ends is once redundant; one compatibility equation gives

$$M_{\text{end}} = \frac{wL^2}{12}\cdot\frac{1}{1 + 2EI/(S_j L)}$$

and the mid-span moment follows from statics as $wL^2/8 - M_{\text{end}}$.

The single dimensionless group in it, $S_j L / EI$, is the whole story: it is the joint’s stiffness measured in units of the beam’s own, which is the same comparison the classification boundaries are built on and the reason they had to be.

Read the two limits. At $S_j \to \infty$ the factor is 1 and the beam is fixed-ended. At $S_j \to 0$ the factor is 0 and the beam is simply supported. Everything between is the curve above, and its shape is worth noticing: it rises steeply at first and then flattens, so most of the redistribution happens at stiffnesses well below the rigid boundary. Half of the fixed-end moment has arrived by $S_j = 2EI/L$, which is a quarter of the way to rigid.

Why it is usually safe #

For the beam, the redistribution is helpful and the direction is easy to see.

The beam was designed for $wL^2/8$ at mid-span. It actually carries $wL^2/8 - M_{\text{end}}$, which is less. A flush end plate takes it down by $0.30 \times wL^2/12 = 0.025wL^2$, which is 20% of the design moment. The beam is over-designed by a fifth without anybody intending it.

The deflection improves too, and by more than proportionally: end restraint reduces mid-span deflection faster than it reduces mid-span moment, because it changes the curvature over the whole span rather than at one point. For a beam that was governed by deflection rather than by strength, which is most long-span beams, that is the more valuable of the two effects.

So the standard practice is conservative for the member it is applied to. That is why it survives.

It is worth being precise about the shape of that conservatism, because “conservative” is doing two different jobs in the sentence. The beam is conservative in strength, having been sized for a moment larger than it carries. It is also conservative in stiffness, having been assumed to deflect more than it does. Both errors point the same way, which is unusual — most modelling simplifications are safe on one and unsafe on the other, and it is the coincidence of the two that makes simple design as robust in practice as it is.

The coincidence is not an accident. Both quantities depend on the same end restraint, and adding restraint reduces both mid-span moment and mid-span deflection simultaneously. What breaks the pattern is not the beam at all. It is everything else the moment reaches on its way out of the beam, which is the next section.

Where it is not safe #

The moment did not disappear. It went into the column.

A column supporting a beam through a “simple” connection is designed for its axial load plus a nominal eccentricity — typically the beam reaction acting at a small distance from the column’s face, which produces a modest moment. What it actually receives is that plus the end moment above, which for a flush end plate is 30% of $wL^2/12$ and can be several times the nominal figure.

That is a beam-column problem the column was not designed as, and the column is the member for which an unexpected moment matters most, because a compression member’s response to a moment is amplified rather than linear. A 20% increase in moment on a column near its buckling load is not a 20% increase in stress.

Three specific hazards follow.

Unbalanced spans. A column with a long beam on one side and a short one on the other receives the difference between two end moments, not their sum. Where both were assumed zero, the difference was assumed zero too — and the difference is the worse quantity, because it is a subtraction of two numbers each of which was set to zero for convenience rather than measured.

A beam on one side only. An external column carries no balancing moment at all, so it receives the whole of whatever its single beam sheds. That is the column with the smallest section on the grid and the largest unbalanced moment on it, which is a combination worth noticing because it is the ordinary arrangement rather than an unusual one.

The end connection’s own capacity. A connection detailed as pinned has a moment capacity, but a small one — web cleats reach a fraction of a beam’s plastic moment. If the joint’s stiffness delivers more moment than its strength can carry, the joint yields and redistributes, which is fine provided it has the rotation capacity to do so. A joint that is stiff and brittle is the bad combination, and it is available: a thick end plate with under-sized bolts is exactly that.

Fatigue. A connection assumed to carry no moment, carrying a fluctuating one, is a detail accumulating damage in a calculation that never included it — and fatigue is governed by the range rather than by the margin.

The reverse error #

The mirror image is at least as common and it is not conservative for the beam.

A frame analysed as rigid, on connections that are semi-rigid, gets the opposite discrepancy — and it is much larger than the first one, for a reason worth seeing.

The analysis predicts an end moment of $wL^2/12 = 0.0833wL^2$ and a mid-span moment of $wL^2/24 = 0.0417wL^2$. An extended end plate delivers 62.2% of the end moment, or $0.0518wL^2$, so the mid-span moment is $wL^2/8 - 0.0518wL^2 = 0.0732wL^2$.

That is 76% more than the analysis predicted, from a connection everybody would call rigid.

The asymmetry between the two errors is arithmetic rather than physical. The simply supported design’s error is measured against $wL^2/8$, which is the largest moment on the beam; the rigid design’s error is measured against $wL^2/24$, which is the smallest. The same absolute shortfall in end moment is a fifth of one and three quarters of the other.

For a beam sized on the rigid analysis, that is a real overstress at mid-span, and it is in the unsafe direction. It is also easy to miss, because everything about the connection looks substantial: an extended end plate with eight bolts and full-strength welds is a connection anybody would call rigid.

The saving grace is moment redistribution: a beam whose support moment falls short and whose mid-span moment overshoots is doing exactly what a plastic analysis would let it do deliberately, and provided the section is Class 1 and the beam has the rotation capacity, the collapse load is unchanged. So the ultimate capacity is usually fine and the serviceability behaviour is not — deflections are larger than predicted, and the beam may sag visibly under load it was told it would not.

Both errors are therefore serviceability problems at the ultimate limit state and strength problems nowhere, which is a strange conclusion and the reason the practice survives. A structure can be wrong about all its joint stiffnesses and still be strong enough, because ductility redistributes what stiffness misallocated.

The deflection, which nobody redistributes #

Strength errors are absorbed by ductility. Deflection errors are not, and that asymmetry is the reason the rigid-analysis error is the more serious of the two in practice.

For the same beam with springs at both ends, the mid-span deflection runs from $5wL^4/384EI$ at $S_j = 0$ to $wL^4/384EI$ at $S_j = \infty$ — a factor of five between the two idealisations. An extended end plate at 62.2% of fixed sits well inside that range, and a beam sized on the rigid analysis will deflect substantially more than the analysis said.

There is no plastic mechanism that recovers a deflection. A beam that has redistributed its moments has the same deflection it had before, or slightly more; the redistribution moved where the material yielded, not how far the beam moved. So a structure that is comfortably strong can still be visibly and unacceptably flexible, which is the distinction the deflection field was built on arriving here through the joints.

The practical consequence is a check ordering. If the beam is deflection-governed — and long-span beams generally are — then the joint stiffness assumption is not a strength approximation at all. It is the input to the governing calculation, and its error goes straight into the answer with nothing to absorb it.

Where the redistribution stops being available #

That last sentence has a condition on it and the condition is the whole risk.

Redistribution needs the over-loaded region to yield and keep carrying while the rest catches up. Everything on this site that relies on it — plastic hinges, block shear, bolt groups at the instantaneous centre — relies on the same property.

Three situations remove it:

• A slender section, where the compression flange buckles locally before the plastic moment is reached, so the beam cannot form a hinge at all;
• A brittle connection component, most often a bolt in tension or a weld, which fractures rather than yielding;
• Fatigue or low-temperature service, where a crack does the redistributing instead of a hinge.

In any of those, an unintended redistribution is not absorbed. It arrives at whichever element cannot yield, and that element decides.

Detailing so that the assumption becomes true #

There is a third option beside “analyse it properly” and “accept the error”, and it is the one most used in practice: detail the connection so that the idealisation is a good one.

For a joint intended to be pinned, that means deliberately making it flexible. The detailing moves are all about removing lever arm and adding rotation capacity:

• connect to the web only and leave the flanges free, which caps the couple at the depth of the cleat rather than the depth of the beam;
• use a thin plate — a fin plate of 8 or 10 mm rather than 15 — because the flexibility of a plate in bending goes as $t^3$ and thinning it is the fastest lever available;
• use long bolt gauges and standard clearance holes, so the connection can rotate a little before anything bears;
• avoid notching or stiffening that inadvertently makes the joint stiff.

For a joint intended to be rigid, the moves are the reverse and are more expensive: extended end plates, column web stiffeners opposite both flanges, thick plates, and welds developing the flange forces.

What is worth noticing is the asymmetry in effort. Making a joint genuinely pinned is nearly free, and making one genuinely rigid is not. A fin plate is cheaper than a flush end plate, which is cheaper than an extended end plate with stiffeners — so the idealisation that costs nothing to achieve is the one the industry defaults to, and the one that costs money is the one most often assumed rather than delivered.

That ordering explains a good deal about which of the two errors shows up in practice. The pinned assumption is usually approximately true because somebody paid nothing to make it so. The rigid assumption is often approximately false because making it true costs stiffeners.

What a designer can actually do #

The honest position is not that every frame should be analysed semi-rigidly. It is that the two idealisations are approximations whose direction of error is known, and the useful discipline is to know which one is being made and check the thing that suffers.

If the beam is designed as simply supported, the check that is missing is on the column. Take the joint’s actual stiffness, compute the end moment, and check the column as a beam-column with it. For most connections detailed as pins the answer is comfortable; for a flush end plate on a light column it may not be.

If the frame is designed as rigid, the check that is missing is the mid-span moment and the deflection. A joint at 62% of fixed-end delivers a mid-span moment 76% above the analysis, and a deflection that no serviceability calculation predicted.

And in both cases the connection’s rotation capacity is the thing that makes the error survivable. It is not usually checked and it is not usually a number anybody has. It is, however, the reason the whole practice works, and a connection detailed so that its weakest component is a plate rather than a bolt has it and one detailed the other way round does not.

What to take from it #

A simply supported design gives the beam a moment it was not designed for at the ends, and gives the column one nobody computed at all. The beam is over-designed by up to a fifth; the column is under-designed by whatever the beam shed.

A rigid design has the opposite error, and it is unsafe at mid-span by far more. The stiffest connection in ordinary use delivers 62% of the fixed-end moment, which is a 76% overshoot at the centre of the span — because the same absolute shortfall is measured against $wL^2/24$ rather than against $wL^2/8$.

Both are usually absorbed by redistribution rather than by margin. That is a statement about ductility, not about conservatism, and the three situations that remove ductility remove the whole argument with it.

The check that is missing is never on the member that was designed. It is on the one at the other end of the joint.

And the deflection has nothing to absorb its error. A structure that got the joint stiffness wrong will still be strong enough, because ductility redistributes; it will not be stiff enough, because nothing redistributes a deflection. For the beams where that matters most — the long ones, the deflection-governed ones — the joint stiffness assumption is not an approximation in a strength calculation at all. It is the input to the governing one.

Detailing to make the assumption true is asymmetric in cost. A genuinely pinned joint is cheaper than a semi-rigid one; a genuinely rigid one is dearer. Which is why, of the two errors in this essay, the one that is expensive to avoid is the one that keeps happening.