Neither pinned nor rigid, which is every real connection

Every frame analysis on this site has assumed one of two things about each joint. Either the members’ ends are free to rotate relative to one another, in which case the joint transmits no moment and is called pinned. Or they rotate together, in which case the joint transmits whatever moment continuity requires and is called rigid.

Both are limits. Every real connection is somewhere between them, and the interesting question is not how far between — it is what the question even means, because the answer turns out to depend on the beam.

A joint is a rotational spring #

Start from the definition. A joint carries a moment $M$ and permits a relative rotation $\phi$ between the members it joins. The relation between the two is the joint’s moment–rotation curve, and everything follows from it.

• A pinned joint has $M = 0$ for any $\phi$: a horizontal line along the rotation axis.
• A rigid joint has $\phi = 0$ for any $M$: a vertical line up the moment axis.
• A real joint has a curve between them: stiff at first, softening as components yield, flattening at its moment capacity.

The three quantities that describe such a curve are its initial stiffness $S_j$, its moment capacity $M_u$, and its rotation capacity — how far it can go before something breaks. Design uses all three for different purposes and they are independent: a joint can be stiff and weak, or flexible and strong.

The curve drawn above is a three-parameter power model,

$$M = \frac{S_j \phi}{\left[1 + \left(S_j\phi/M_u\right)^N\right]^{1/N}}$$

which is asymptotic to $S_j$ at the origin and to $M_u$ at infinity, with $N$ setting how sharply it turns. Nothing about it is derived; it is a shape that fits measured joints. What it is used for here is the question of where a joint sits between two bounds, and that question is insensitive to the exact curvature.

The boundaries are not on the joint #

Here is the result that makes the classification counterintuitive, and it is worth stating before any numbers.

A joint is called rigid if it is stiff enough that assuming full continuity introduces acceptable error. Whether a given stiffness is “enough” depends on what it is being compared with — and the thing it is compared with is the flexibility of the beam it is holding.

A very stiff beam is hard to bend, so a joint of moderate stiffness looks flexible beside it. A very flexible beam is easy to bend, so the same joint looks nearly rigid. The comparison is between $S_j$ and $EI/L$, and the boundaries are therefore multiples of $EI/L$:

$$S_j \ge \frac{8EI}{L}\ \text{(braced)} \quad\text{or}\quad \frac{25EI}{L}\ \text{(unbraced)} \qquad\Longrightarrow\qquad \text{rigid}$$

$$S_j \le \frac{0.5EI}{L} \qquad\Longrightarrow\qquad \text{pinned}$$

For the 6 m beam above, $EI/L = 14{,}000$ kN·m/rad, so the rigid boundary is 112,000 and the pinned boundary 7,000. Both drawn on the figure as rays, because a stiffness is a slope rather than a level.

Two multipliers for the braced and unbraced cases, and the factor of three between them is not arbitrary. In a braced frame the joint’s flexibility affects the beam’s moment distribution and nothing else. In an unbraced frame it also affects the frame’s sway stiffness, which affects its second-order amplification, which affects everything. So a joint has to be three times stiffer to be treated as rigid when the frame depends on it to stand up.

What the three connections physically are #

The three curves are worth a sentence each, because the enormous spread between them — a factor of twenty-five in stiffness — comes from geometry rather than from quantity of steel.

Web cleats. Two angles bolted to the beam’s web and to the column. The beam’s flanges are not connected to anything. To transmit a moment, the couple would have to be carried across a lever arm no larger than the depth of the cleats, and the angles themselves bend easily — so the joint is flexible almost by construction. It is the joint that most nearly achieves what “pinned” means, and it still delivers 6% of the fixed-end moment.

Flush end plate. A plate welded across the beam’s end, bolted to the column, with all the bolts inside the beam’s depth. Now the moment is a couple between bolt rows near the top flange and bearing at the bottom flange, and the lever arm is most of the beam’s depth. That is a large increase over the cleats, and it costs one plate.

Extended end plate. The same plate, projecting above the top flange, with an extra bolt row outside the section. That row is further from the compression centre than any inside row can be, so it contributes more to both the stiffness and the capacity than its own count suggests.

The progression is the second moment of area argument once more, and it is now appearing in its fifth setting on this site. The stiffness of a moment connection goes as the square of the lever arm — see the $z^2$ in the component method — so pushing a bolt row further from the compression centre buys quadratically, while adding a row nearer the middle buys almost nothing.

That is why an extended end plate is nearly four times as stiff as a flush one for a modest amount of extra plate, and it is why the whole design instinct in moment connections is to reach further apart rather than to add more.

Where real connections land #

The three joints in the figure are ordinary, and the arithmetic against that 6 m beam is unflattering.

The last column is the end moment as a fraction of the fixed-ended value $wL^2/12$, and it is the number that matters for the analysis — because a beam analysed as fixed-ended and connected with an extended end plate has 62% of the end moment the analysis predicted, and correspondingly more at mid-span.

The extended end plate is about as stiff as ordinary steelwork gets. It reaches 41% of the rigid boundary. A connection that anybody would describe without hesitation as a rigid connection is delivering under two-thirds of the continuity the word implies.

That is not a scandal, and the practice of designing such frames as rigid is not wrong. It is a statement about where the conservatism lives, and it is worth knowing which direction it points, which is the subject of its own essay.

Why the same joint changes class #

The consequence of the boundaries being multiples of $EI/L$ is that the classification moves with the beam, and it moves further than intuition allows.

Take the flush end plate at 12,000 kN·m/rad and give it three different beams:

• a 6 m beam of $EI = 84{,}000$: $EI/L = 14{,}000$, rigid boundary 112,000 — the joint is semi-rigid at 11% of it;
• a 12 m beam of the same section: $EI/L = 7{,}000$, rigid boundary 56,000 — still semi-rigid, but now at 21%;
• a 12 m beam of a section a quarter as stiff: $EI/L = 1{,}750$, rigid boundary 14,000 — the same joint is now at 86% of rigid and very nearly qualifies.

Nothing about the connection changed in any of those. Same plate, same bolts, same welds, same $S_j$. What changed is what it is holding.

The general statement is that connection classification is a property of the joint and the member together, and treating it as a property of the connection alone is the error the boundaries are written to prevent. It is also the reason a connection cannot be classified on a standard detail sheet — the sheet does not know what beam it will be used on.

Where the boundary numbers come from #

The multipliers 8, 25 and 0.5 look like committee numbers and two of them can be derived, which is worth doing because it says what “acceptable error” was taken to mean.

Take a uniformly loaded beam with equal rotational springs at both ends. Solving it — one redundancy, one compatibility equation — gives an end moment of

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

Set $S_j = 8EI/L$ and the factor is $1/(1+0.25) = 0.80$. So the rigid boundary is the stiffness at which a joint delivers 80% of the fixed-ended moment — an error of 20% in the end moment, accepted because the corresponding error at mid-span is smaller and in the safe direction.

Set $S_j = 0.5EI/L$ and the factor is $1/(1+4) = 0.20$. The pinned boundary is where a joint delivers 20%, which is the mirror image: a beam designed as simply supported with 20% of a fixed-end moment at its ends has a mid-span moment 10% below what it was designed for, which is conservative.

So the two boundaries are symmetric statements about a 20% modelling error, and the whole classification is one decision — how much error to tolerate — expressed twice.

The 25 for unbraced frames does not come out of that calculation, because the quantity it protects is the frame’s sway stiffness rather than the beam’s moment. It comes from requiring the frame’s second-order amplification to be computed within a similar tolerance, which turns out to need a much better joint because amplification is a divergent series and errors in its denominator do not stay small.

What this does not settle #

Three limitations are worth recording, because the classification is often read as settling more than it does.

Classification is about stiffness alone. It says nothing about the joint’s moment capacity or its rotation capacity, and a joint can be stiff and weak. Classifying a joint as rigid licenses an analysis assumption; it does not license the moment that analysis then produces.

The initial stiffness is not the operating stiffness. $S_j$ is the tangent at the origin, and a joint working at half its moment capacity is well past that. Design practice uses a reduced secant stiffness — commonly $S_j/2$ or $S_j/3$ — for the analysis, and the reduction pushes joints down a class.

The curve is not repeatable in the way a material property is. It depends on bolt preload, on plate flatness, on weld profile and on whether the joint has been loaded before. Two nominally identical connections can differ by a factor that would be shocking in a material test, which is why the classification bands are wide.

The third idealisation, which is the honest one #

There is an option the two-way choice hides, and it has been available in codes for thirty years: analyse the frame with the joints as springs, at their actual stiffness.

That is what a semi-rigid analysis is, and mechanically it is not difficult — a rotational spring at each member end is two extra degrees of freedom and any frame program can do it. The reasons it is rare are practical and they are worth knowing because they are not reasons of principle.

The joint has to be designed before the frame can be analysed. A rigid or pinned analysis needs nothing from the connection; a semi-rigid one needs $S_j$, which needs the plate thickness, the bolt layout and the column section. So the design sequence inverts, and the usual iteration — analyse, size members, design connections — becomes a loop.

The stiffness has to be a number somebody will stand behind. $S_j$ from the component method is a calculation with a real scatter, and using it in the analysis propagates that scatter into the member forces.

And the answer is not obviously better. A semi-rigid analysis of a braced frame produces end moments the simple design ignored and mid-span moments smaller than it assumed. Whether that is worth having depends on whether the beam was governed by the mid-span moment, and often it was not — it was governed by deflection, which semi-rigid analysis improves only slightly.

Where it does pay is the case its proponents have always pointed at: braced frames where the beams are deflection-governed and long. There the end restraint a “pinned” connection actually supplies reduces the mid-span deflection substantially, and taking credit for it saves steel that the simple model throws away. That is a real saving and it is the reason the method exists; it is also a narrow enough case that most designers never meet it.

Where the stiffness comes from #

Nothing above says why a flush end plate has a stiffness of 12,000 and an extended one 46,000. That is a question about what the joint is made of, and the answer is that it is made of springs in series — the column web, the column flange, the end plate and the bolts, each with its own flexibility, all adding.

What to take from it #

Every joint is a rotational spring, and pinned and rigid are its two limits. Real connections sit between them and the position is measurable.

The classification boundaries are multiples of the beam’s $EI/L$. The same connection is rigid on one beam and semi-rigid on another, and the classification cannot be made without naming the member.

The stiffest connection in ordinary use reaches 41% of the rigid boundary for an ordinary beam. “Rigid connection” is a description of intent rather than of behaviour.

And the braced and unbraced boundaries differ by a factor of three, for a reason. In an unbraced frame the joint’s flexibility feeds into the sway stiffness and therefore into the second-order amplification, so it has to be three times better to be ignored.

One more, which is the practical one. The whole of the spread from 1,800 to 46,000 kN·m/rad — a factor of twenty-five between the softest and the stiffest connection here — was bought with lever arm rather than with material. Web cleats to a flush end plate is one plate; a flush end plate to an extended one is a strip of that plate projecting above the flange, and it is worth a further factor of 3.8. The connection field turns out to obey the same rule the section field established at the beginning of this site, and the rule is that where the material sits decides more than how much of it there is.