A joint made of springs in series

The last essay established that a joint’s rotational stiffness decides what a frame analysis is allowed to assume, and left one question unanswered: where does the number come from?

The answer is that it is not a number about the connection. It is a number about everything the force passes through on its way from one member to the other, and those things are in series.

Series, and what series means #

Follow the tension from the beam’s top flange to the column’s web. It passes through, in order:

1. the end plate, which bends;
2. the bolts, which stretch;
3. the column flange, which bends;
4. the column web in tension, which stretches;

and on the compression side, through the beam flange, the column web in compression which squashes, and the column web panel which shears.

Every one of those deforms. The rotations they produce add, because they happen one after another along the load path, which means their flexibilities add:

$$\frac{1}{k_{\text{eq}}} = \sum_i \frac{1}{k_i}$$

That is the series rule and it is the whole of the method’s structure. The rotational stiffness follows as

$$S_j = \frac{E z^2 k_{\text{eq}}}{1}$$

with $z$ the lever arm between the tension and compression centres — the $z^2$ being the same quadratic that makes material far from the neutral axis do nearly all the work in a section.

The softest one decides #

A series combination has an unavoidable property: it is always softer than its softest element, and it is dominated by it.

For the joint above the flexibilities are:

The column flange contributes nearly two-fifths of the joint’s total flexibility. The bolts contribute under a tenth.

And the consequence for design is the arithmetic in the right-hand column of the figure. Double each component’s stiffness in turn and see what the joint gains:

Doubling the bolts buys 4.6%. Doubling the thing everybody would think of first — the end plate — buys 9.9%. The only intervention worth making is on the column flange, and even that buys under 25% for twice the stiffness.

That is the practical content of the whole method, and it is a result about arrangement rather than about steel. A chain of springs cannot be improved by working on the stiff links.

Two of the five are not in the connection #

Look at the list again and notice where the components live.

The column web in shear, and the column web in compression, are parts of the column. Between them they are 33.5% of the joint’s flexibility, and neither of them appears on the connection drawing.

Which means a connection detail does not have a stiffness. The same end plate, the same bolts and the same welds on a heavier column produce a stiffer joint, and on a lighter one a softer one — and the difference is a third of the total.

That is why the classification of the previous essay cannot be printed on a standard detail sheet, and it is a second reason on top of the one that essay gave. The boundaries depend on the beam; the stiffness depends on the column; and the connection detail is only the middle third of the answer.

It has one further consequence that is easy to meet in practice before meeting it in theory. A connection that was designed and checked for one frame, and then reused on a job where the columns are lighter, is not the same joint — and nothing on the detail changed to say so. Reuse of connection details across a project is standard and sensible practice; reuse across projects with different column sections silently moves a third of the joint’s flexibility.

It also explains a common and otherwise puzzling detail: column web stiffeners. Two plates welded into the column opposite the beam’s flanges do nothing for the bolts, nothing for the end plate, and nothing that appears on any bolt schedule. What they do is remove two of the five springs from the chain by making the column web locally rigid — which is the largest single intervention available, precisely because the components they remove are large ones.

Where the numbers in the table come from #

The stiffness coefficients above are dimensionless — 2.1, 3.8, 4.6 — rather than stiffnesses in kilonewtons per millimetre, and the reason is a factoring that makes the whole method tractable.

Each component’s actual stiffness is written as $E k_i$, with $E$ the modulus taken out in front and $k_i$ carrying the geometry. So $k_i$ for a bolt is $A_s/L_b$, an area over a length; for a plate in bending it is a function of $t^3$ over $m^3$ times a width. Every one is a length, and every one is pure geometry.

Two things follow, and both are more useful than they look.

The modulus cancels out of every comparison. The shares in the table — 39.6% for the column flange, 8.9% for the bolts — are properties of the geometry alone, and they would be identical in aluminium. That is the one number a stronger steel does not change doing structural work: because every component shares the same $E$, a material change scales the whole joint and reorders nothing.

The decomposition is stable across grades and sizes. A designer who learns that the column flange dominates an ordinary end-plate joint has learned something that transfers, rather than a fact about one calculation.

The exception is a joint made of two materials — a steel plate bolted to a timber member, a steel base plate on concrete — where the moduli differ by a factor of twenty or more. There the factoring fails, the softest component is whichever one is in the weaker material, and it is not close.

Where each component’s stiffness comes from #

The method is only useful if each $k_i$ can be computed rather than looked up, and each of them can. Two examples show the range.

Bolts in tension. A bolt is a bar of area $A_s$ and effective length $L_b$, so its stiffness is $EA_s/L_b$. The only subtlety is $L_b$, which is not the bolt’s length but the clamped length plus allowances for the head and nut. This is the easiest component and, as the table shows, the least important.

The end plate in bending. This is the prying problem again, seen from the stiffness side rather than the strength side. The plate is a tee stub; its flexibility depends on $m$, on the bolt pitch and on $t^3$. The cube is why plate thickness is such an effective lever on stiffness and why it appears in a different power than in the strength check, where it is $t^2$.

That difference of power is worth pausing on. Increasing the plate thickness by 20% raises its strength contribution by 44% and its stiffness contribution by 73%. A connection detailed to be strong enough is usually stiffer than the calculation that sized it — which is a rare instance of two design requirements pulling the same way.

The column flange in bending is the same tee stub calculation applied to the column, and its dominance in the table is a geometric accident of ordinary sections: a column flange is thinner relative to its span between the web and the bolts than an end plate is, so it bends more.

The same decomposition, for strength #

Everything above is about stiffness, and the method’s other half does strength — with one structural difference that is worth stating because it inverts the design guidance.

For stiffness the components are in series, so the flexibilities add and the softest dominates.

For strength the components are also in series in the sense that the force passes through all of them, but the consequence is different: the capacity is the minimum rather than a sum of reciprocals. Whichever component reaches its limit first sets the row’s resistance, and everything else has capacity to spare.

So the same decomposition answers two questions and the answers name different components. It is entirely ordinary for a joint’s stiffness to be dominated by the column flange while its strength is limited by the column web in compression — and then stiffening the flange and strengthening the web are two different jobs on two different parts.

There is a third question that the same decomposition answers and that is easier to forget. Which component reaches its limit first also decides whether the joint is ductile, because the components are not equally ductile. A plate in bending has a great deal of deformation available past its yield; a bolt in tension has very little. A joint whose weakest component is the end plate deforms visibly and redistributes; one whose weakest component is the bolts fractures.

That is the reason design guidance prefers connections where a plate governs, and it is the same preference the bearing essay recorded — make the ductile mechanism the weakest — arriving through a completely different route.

Why the method is worth its arithmetic #

The component method replaces a table of connection types with a calculation, and the reasons it is worth the trouble are not the obvious one.

It is compositional. A joint configuration nobody has tested can be assessed by decomposing it into components that have been. That is how a method survives contact with the geometry a real building produces.

It says where to intervene. The table above is not available from any tabulated stiffness, and it is the thing a designer actually needs — not “this joint is 25,228 kN·m/rad” but “and the only way to change that is the column flange”.

It reuses one calculation everywhere. The tee stub appears three times in the five components, in different orientations. The same is true of the strength side, where the end plate’s bolt rows are tee stubs and the column flange opposite each one is another.

The lever arm, which is outside the series entirely #

There is one quantity in $S_j = E z^2 k_{\text{eq}}$ that is not a spring at all, and it is the one with the largest exponent.

$z$ is the distance between the tension centre and the compression centre — roughly the beam’s depth for a flush end plate, and more than that for an extended one. It multiplies the stiffness quadratically, and it is not in the series chain, so nothing about the softest component limits what it can buy.

Compare the two levers honestly. Doubling the best component in the chain buys a factor of 1.25. Increasing $z$ by 25% buys a factor of 1.56, and increasing it by 50% buys 2.25.

The lever arm beats every component in the series, by a wide margin, and it is the one variable a series argument cannot dilute. That is why the progression from web cleats to flush end plate to extended end plate in the previous essay produced a factor of twenty-five: those three connections differ mostly in $z$.

It is also why deepening the beam is such an effective way to stiffen a frame, and why a shallow beam with a heavily engineered connection is usually a worse arrangement than a deeper beam with a simple one. The same conclusion arrived at in the deflection field for members holds for the joints between them, and for the same reason.

Where the model is thinner than it looks #

Two honest limitations.

Series is an idealisation. The components are not truly independent: the column flange’s deformation changes the bolt’s effective length, and the end plate’s contact with the column flange is what generates the prying force in both. The method treats them as separable because the alternative is a contact problem in three dimensions, and it is validated against tests rather than derived.

Initial stiffness is a tangent. All of the above computes $S_j$ at the origin, and a joint working near its capacity is much softer — which is why analysis uses a reduced value, commonly $S_j/2$ or $S_j/3$, and why a joint that classified as rigid on its initial stiffness may not on its operating one.

What to take from it #

A joint’s stiffness is a series combination, so the flexibilities add and the softest component decides. The arithmetic is unforgiving: four of five components here return under 13% for twice the stiffness.

A third of the flexibility is in the column, not the connection. Which is why a connection detail has no stiffness of its own, and why column stiffeners are the largest single intervention available.

The plate thickness enters strength as $t^2$ and stiffness as $t^3$. A joint detailed to be strong is usually stiffer than it needed to be, which is not the usual relationship between two design requirements.

And the method’s real product is the table of gains, not the stiffness. Knowing a joint is 25,228 kN·m/rad is worth much less than knowing that only one of its five components is worth touching.

The lever arm is outside the series and beats everything in it. $S_j$ goes as $z^2$, so a 50% increase in the distance between the tension and compression centres is worth 2.25 — more than doubling any two components together. Every component in the chain competes with the others for a share of one flexibility; $z$ does not compete with anything.

There is a general shape here that reaches past joints. A series arrangement hides its own leverage: it makes every element look equally important on the drawing and makes them wildly unequal in the arithmetic, and the only way to tell which is which is to compute the shares. The same is true of a load path through a structure, of a chain of tolerances through a fabrication, and of the sequence of checks a connection has to pass. What the component method really supplies is the habit of asking, before improving anything, what fraction of the total it is.