How much of the plate is bending

The tee stub is the unit every moment connection is assembled from, and it is an object nobody builds. What gets built is a rectangle of steel with several bolts through it, welded to a beam along two lines, restrained by a column flange on the other side and free at its edges. The whole component method rests on being able to replace that with a tee — and the replacement needs one number.

The number is a length, and it is not a dimension anybody can measure on the plate.

What a yield line is doing here #

A plate that has yielded does not yield everywhere. It yields along lines, and the lines are hinges: straight or curved creases across which the plate rotates while the panels between them stay flat.

A collapse mechanism is a set of such lines that lets the plate move. Its resistance is the plastic moment per unit length of hinge — $\tfrac14 t^2 f_y$ — multiplied by the total length of hinge, divided by the movement the mechanism needs. A longer pattern is a stronger one, and the plate collapses along the shortest pattern it can find, which is exactly the argument a plastic collapse mechanism makes about a frame with hinges in members instead of lines in plate.

So the search is a minimisation, and its answer has units of length. That length — with the $\tfrac14 t^2 f_y$ divided back out — is the effective length, and it is defined as the length of tee stub that would collapse at the same load.

Two families, and only one of them is a circle #

Two shapes of pattern compete, and the competition is the interesting part.

The circle. A cone of hinges closing round the bolt, with the plate folding into a shallow dish. Its length is $2\pi m$, where $m$ is the distance from the bolt to the web face. It needs nothing from the plate’s edges — it is entirely local to the bolt.

The fan. Straight hinges running from beside the bolt out to the free edges of the plate. Its length is $4m + 1.25e$, where $e$ is the distance from the bolt to the edge. It reaches the boundary and its length depends on where the boundary is.

The plate takes whichever is shorter. On the plate above that is the fan, at 249 mm against 283.

That crossing is worth holding, because it says something a designer can use without any arithmetic. A bolt close to the web collapses the plate in a circle; a bolt far from it collapses the plate in a fan. The boundary is at about half the edge distance, and it moves the whole calculation from one that depends on the plate’s edges to one that does not.

The rule that produces two lengths at once #

Here is where the calculation stops being a single minimisation, and it is the part that surprises everybody the first time.

The three collapse modes a tee stub has are the flange folding on its own, the flange and the bolt failing together with prying, and the bolt failing alone — the three regimes the rung below is about.

The circular pattern can only form in the first of them.

The reason is geometric rather than a convention. A circle of hinges round a bolt is a fold that goes all the way round; for the plate to complete that fold, the bolt has to be there throughout, holding the middle down. In a mode where the bolt breaks, the bolt is not there, and the fold has nowhere to close against — so the plate must find a pattern that reaches its own free edge instead.

So one calculation carries two effective lengths. Mode 1 uses the smaller of the two families. Modes 2 and 3 use the straight family alone, however much longer it is. On the plate above the two happen to coincide; move the bolt in to 25 mm and they separate — 157 mm for mode 1 and 169 mm for mode 2, from the same plate at the same instant.

What the three modes then do #

Read the crossings rather than the curves. A plate thinner than 14.4 mm collapses by folding and the bolts are never in danger. A plate thicker than 24 breaks its bolts and the plate is irrelevant. Between the two, both happen and the prying force is the difference between them.

The design consequence is the one this ladder keeps arriving at from different directions: the plate thickness chooses the failure mode, and the two modes it chooses between fail in completely different ways — one with a great deal of warning and one with none.

What the arithmetic looks like written out #

The three modes are three formulas and they are short enough to carry.

Mode 1, the flange folding on its own, is a mechanism with hinges at the web and at the bolt line:

$$F_1 = \frac{4 M_{pl,1}}{m}, \qquad M_{pl,1} = \tfrac14 \, \ell_{\text{eff},1} \, t^2 f_y$$

Mode 3 is the bolt row’s own resistance and has no geometry in it at all.

Mode 2 is the two together, and it is the one with prying in it:

$$F_2 = \frac{2 M_{pl,2} + n \sum B}{m + n}$$

where $n$ is the distance out to the prying reaction, taken as the smaller of the edge distance and $1.25m$ — a cap that exists because a prying force cannot act beyond the point at which the flange has lifted off.

Every geometric quantity in those three appears exactly once, and the effective length appears in two of them at two different values. That is the whole calculation; everything else in this essay is about how the two values of $\ell_{\text{eff}}$ were arrived at.

There is a check worth doing on any of it, and it takes a second. Multiply the effective length by the thickness squared and by a quarter of the yield stress, and the answer has to be a plastic moment in the right range for a plate of that size. On the inner row above: $\tfrac14 \times 200 \times 15^2 \times 275 = 3.09$ kNm, over a lever arm of 45 mm, four hinges — 275 kN, which is what the figure reports for mode 1. A calculation that has lost a factor somewhere fails that arithmetic loudly.

The same bolt in three places #

Which brings the essay to its own finding, and it is a large number.

One bolt row, one plate, one thickness, three positions: 255, 226 and 137 kilonewtons. The bolt is identical in all three and its own capacity — 352 kN — is reached in none of them.

That spread is the reason the component method exists. A designer working from bolt capacities would size all three the same; the plate they are in decides otherwise, and it decides by a factor of nearly two.

What moving the bolt actually buys #

The last figure’s crossing suggests an obvious move, and the obvious move only half works.

Bringing the bolt closer to the web shortens the lever arm on the flange, which raises the mode 1 resistance as $1/m$. It also shortens the yield-line pattern, which lowers it in proportion to the length. The two go opposite ways.

Move the inner row’s bolt from 45 mm to 25 and the row goes from 255 kN to 288 — a gain of 13 per cent, where the lever arm alone would have promised eighty. The effective length fell from 200 mm to 157 on the way, and the pattern changed family: at 45 mm the plate folds in a fan and at 25 it folds in a circle.

There is a limit to the move that has nothing to do with any of this. A bolt has to be far enough from the web for a spanner to reach it, and that clearance — a number from a fabrication handbook — is what actually sets $m$ on most connections. The strongest position is usually not available, which is the ordinary condition of connection design and is worth saying because the arithmetic above makes it look like a free variable.

Why any of this is worth the trouble #

There is a simpler way to size an end plate and it is worth saying why it is not used.

The simple way is to take each bolt at its own capacity and make the plate thick enough that it obviously does not bend — a rule of thumb ties the plate thickness to the bolt diameter and stops. It is safe, it is quick, and it produces the connection that fails in mode 3, with no warning and no rotation.

What the yield-line calculation buys is the other two modes, and the reason to want them is not economy.

A joint that fails by folding its plate has rotation capacity, and one that fails by breaking its bolts does not. Everything a frame does after first yield — redistributing moment away from a hinge, reaching a plastic collapse load, surviving a support settlement — requires its joints to turn while holding their strength. A mode 1 joint turns. A mode 3 joint reaches its strength and lets go.

So the effective length calculation is not primarily a strength calculation. It is how a designer finds out which failure they have specified, and the answer is decided by a plate thickness that is otherwise chosen for handling and welding. That is why the mode boundaries in the figures above are the numbers to carry: 14.4 mm and 24 mm on this connection, from a plate that would ordinarily have been drawn at 20 without a thought.

A row is rarely alone, and the plate is not the only thing bending #

A row is not always alone. Two bolt rows close together do not each get their own pattern; they fold as a group, and the group’s effective length is less than the sum of the individual ones. The code handles it with a set of group patterns and a rule that takes the smaller of the individual and the group answers, and the arithmetic above treats every row as isolated.

The plate is not the only thing bending. The column flange on the other side of the bolts is a second tee stub with its own yield lines, its own effective length and its own three modes, and the two are springs in series. The row’s resistance is the smaller of the two and its stiffness is the two combined, so a calculation that stops at the end plate has done half the work.

Yield-line analysis is an upper bound. Every pattern examined gives a resistance, the true collapse load is the lowest of all possible patterns, and the ones tabulated are the ones somebody thought of. A pattern nobody has drawn would give a lower answer, which is why the method is calibrated against tests rather than trusted on its own — and it is the opposite failure direction from a lower-bound method, which is safe by construction and may be wasteful.

And the weld is assumed to be stronger than everything. The distance $m$ is measured from the web face less 0.8 of the weld leg, which is an allowance for the fillet stiffening the plate near the web. It is a correction of a few millimetres to a dimension the answer is inversely proportional to, and it is the sort of allowance that gets copied between calculations without being checked against the weld actually specified.

The measurement that made the method credible #

None of the above would be worth much on its own, because a yield-line analysis is an upper bound and an upper bound is a claim that has to be beaten down by evidence.

The evidence is a large programme of tee-stub tests carried out through the 1970s and 1980s, mostly in the Netherlands and Germany, on specimens whose only variables were the plate thickness and the bolt position. What those tests established is not that the patterns are right — a test cannot see a yield line — but that the three-mode structure is right: that specimens sort cleanly into ones that folded, ones that broke bolts, and ones that did both, and that the boundaries fall where the arithmetic puts them.

Two things about that programme are worth carrying.

It was a test of a component, not of a connection. The specimens were tee stubs, deliberately, so that the plate’s behaviour could be measured without a beam, a column or a moment confusing it. That is the whole idea the component method is named for, and it is why the method transferred to connection types nobody had tested: a component whose behaviour is known can be assembled into an arrangement that is not.

And the calibration is in the effective lengths rather than in a factor. Where a pattern disagreed with the tests, the pattern was changed — new patterns were added to the tables, and the α chart for the row below a tension flange is an entirely empirical curve sitting among the derived ones. The method looks like a piece of plasticity theory and is in part a fitted table wearing plasticity’s clothes, which is worth knowing before treating any single effective length as exact.

A yield line is a band, and the plate does not stay flat #

The yield lines are drawn as lines and they are bands. A plastic hinge in a plate has a width of a few times the thickness, over which the curvature is spread, and the sharp creases in every figure above are an idealisation of a smooth fold. It matters for the deformation the plate needs to reach the mechanism — which is what a joint’s rotation capacity is about — and not at all for its resistance.

The other absence is out of the page. The pattern is drawn on a flat plate and the plate is not flat when it collapses, and the dish it folds into is what the bolt is being pulled through. The membrane action in that dish is a real and unclaimed reserve, ignored in every code because it needs deformations well past anything a connection is allowed to reach.

The assumption underneath the substitution #

The whole method assumes that a plate and a tee stub of the right length fail in the same way, and they do not quite.

A tee stub’s flange bends in one direction, in cylindrical curvature, with hinges parallel to the web. A plate folds in two directions at once, and the circular pattern in particular is a genuinely doubly-curved collapse. The equivalence is one of collapse load only — the two objects reach their limits together, and they get there through different shapes and at different deflections.

That is enough for a strength check and it is not enough for a stiffness one. The component method needs both, and the stiffness of a bolt row is given by a separate formula fitted to tests rather than derived from any of this — which is the honest position, and it is worth knowing that the two halves of the same component come from two different kinds of argument.

The patterns that are still a table #

Later rungs on this anchor: the group patterns, where two rows fold together and the effective length is shared. The α chart for a row adjacent to the tension flange, which is a fitted curve rather than a pattern and is the one place in the method where a number comes from a graph. The column flange as the other tee stub, and how the two combine into one row resistance. Rotation capacity, which is what a plastic distribution across the rows is spending. And the assembly itself, where each row’s resistance and stiffness go into a joint’s own moment–rotation curve.

The tee stub was chosen as the unit because it is the simplest object that shows prying, and everything above is the price of that choice: a real plate has to be converted into one, and the conversion is a plastic collapse analysis carrying two answers, one per mode. The component method’s economy is real, and it is not free — what it buys is that the tee’s arithmetic is then done once and applied everywhere, and what it costs is a table of patterns that has to be right.