The column made of two columns

A solid column has one way to bend, and Euler counted it. A column made of two chords joined by a lattice has two: it can bend, as a solid column does, and the lattice can shear while the chords stay straight. The second mode is not available to a solid member at all, and it is what makes a built-up column a different object rather than a lighter one.

The two stiffnesses are in series #

Engesser’s result is one line, and everything on this page is a reading of it:

$$\frac{1}{P_{cr}} = \frac{1}{P_E} + \frac{1}{S_v}$$

where $P_E = \pi^2 EI/L^2$ uses the second moment of the whole built-up section, and $S_v$ is the shear stiffness of the lattice — the transverse force per unit of shear strain, which for a lacing system is a truss-deflection problem and for battens is a frame one.

That is a harmonic sum, so the smaller term governs and the larger one is nearly irrelevant. Two flexibilities in series, exactly as a beam and its supports are in series and for the same reason: the two mechanisms happen one after the other along the same load path, so their displacements add.

For the column drawn, $P_E = 1{,}341$ kN and $S_v = 31{,}500$ kN, so $P_{cr} = 1{,}287$ kN — 96% of Euler. A four per cent penalty, which is why laced columns are used at all.

Battens, and a factor of four #

The battened column is the finding. Its nominal slenderness is 79 — a perfectly ordinary column — and it behaves like one of slenderness 163, which is at the far end of what any code permits. The chords have not changed, the spacing has not changed, and the whole of the difference is that a batten resists racking by bending while a diagonal resists it by stretching — which is the deflection that is not bending arriving in a place nobody expected it.

A diagonal is an order of magnitude stiffer than a bent member of the same size, which is the argument for triangulating anything and is exactly why a Vierendeel girder is expensive. A battened column is a Vierendeel girder stood on end and asked to be a column.

Which free body produced the shear stiffness #

The lacing’s shear stiffness is a truss deflection, computed the way this collection computes truss deflections.

Take one panel of the lattice, of length $a$, with the chords at spacing $h$. Apply a transverse shear $V$ across it. In a double-laced panel each diagonal of length $d$ carries $V d/(2h)$ — resolved from the shear — and stretches by $Vd^2/(2hEA_d)$. That extension lets the panel rack by an angle, and assembling the geometry gives

$$S_v = \frac{n E A_d\, a\, h^2}{d^3}$$

with $n$ the number of lacing planes. Every term is a length or an area of the lattice, and not one of them is a property of the chords.

That is the unit-load calculation this collection uses for which member moved the roof, applied to one panel and read as a stiffness rather than as a deflection — and the diagonals dominate it here for exactly the reason they dominate a truss’s deflection.

For battens the same panel is a portal frame: the chords bend in double curvature between battens and the battens bend with them, giving

$$S_v = \frac{24 E I_{ch}}{a^2\left(1 + \dfrac{2 I_{ch} a}{n I_b h}\right)} \;\le\; \frac{2\pi^2 E I_{ch}}{a^2}$$

The batten spacing $a$ appears squared in the denominator, so halving the batten spacing quadruples the shear stiffness — which is the only handle a battened column really has, and the reason batten spacings look absurdly close on old drawings. The same three lattices redrawn with the battens at 600 mm instead of 1,200 say how much that handle is worth:

Moving the chords apart, twice #

Here is the result that makes the subject worth an essay, and it is the one place where the obvious move stops working. Every instinct about a built-up column says that the way to make it stronger is to push the chords further apart, because that is what the second moment of area rewards. Sweeping the spacing and reading the critical load at each point says otherwise, and it says it twice, in two different ways depending on what is held constant while the spacing changes.

$P_E$ grows as $h^2$, because the radius of gyration is essentially half the spacing and the Euler load goes as its square. So far so obvious.

$S_v$ does not grow at all. At a fixed lacing angle every length in the lattice scales together — $a \propto h$ and $d \propto h$ — so $S_v \propto h \cdot h^2 / h^3$, which is a constant. The shear stiffness of a lacing system at a fixed angle is scale-free.

The harmonic sum of a term rising without limit and a term that does not move is a curve that approaches the fixed one. So:

$$P_{cr} \to S_v \quad\text{as}\quad h \to \infty$$

A built-up column cannot be made stronger than the shear stiffness of its own lattice, however far apart its chords are put. At 300 mm the column is at 96% of Euler; at 900 mm it is at 73%; at 3 m it is at 20% and has spent almost all of what widening will ever buy. The spacing at which the two stiffnesses are equal — the knee — is 1,478 mm.

Which of the two sweeps applies is decided by what fixes the panel. If the lacing angle is held — the usual detailing rule, 45° to 60° — the ceiling applies. If the panel length is set by something else, a connection spacing, a plate size, a floor level, then the second curve applies and there is an optimum spacing to be found.

Both curves say the same thing in different words: past some spacing, the lattice is the column.

What buys the shear stiffness #

$S_v \propto A_d$, so the lattice’s own area is the direct handle:

The curve is the familiar shape of a series combination — most of the benefit early, and a long expensive tail. Going from 25 to 100 mm² of lacing buys 18 points; going from 400 to 800 buys one. Lacing is sized generously because it is cheap and because the first hundred square millimetres do nearly all the work.

It is the same shape as the bracing problem, and for the same reason: a brace needs a threshold stiffness and nothing beyond it, because once the secondary system is much stiffer than the primary one the primary one is all that is left to fail. Lacing is bracing distributed along a member instead of applied at a point, and its curve flattens where a brace’s curve reaches its plateau.

The ends still decide the length #

Nothing on this page displaces the older argument. $P_E$ in Engesser’s expression is the Euler load at the member’s effective length, not at its geometric one, so a built-up column with poor end restraint suffers the usual factor of up to sixteen before any of the lattice arithmetic begins — and the lattice penalty is then applied on top of it. The ends still decide the length; the lattice only decides how much of what is left survives.

That conversion is how codes handle it. Rather than carrying a shear stiffness through every subsequent calculation, the loss is expressed as an increased slenderness — 79 becomes 163 for the battened column — and everything downstream is the column curve as usual.

What that costs is the currency this collection has used since the beginning. Capacity goes as the inverse square of the effective length, so doubling an effective slenderness quarters the capacity — which is the same arithmetic as making the column twice as long, and here it has been bought with a batten detail rather than with a metre of steel.

Why anybody builds one #

The whole of the above is a catalogue of penalties, and built-up columns are nevertheless everywhere. The reason is in the numerator of the ratio rather than in the fraction.

A rolled section’s radius of gyration is bounded by its own depth, and its depth is bounded by what a mill will roll. The largest universal column has a radius of gyration of about 100 mm about its weak axis. Two of those chords put a metre apart have a radius of gyration of 500 mm — five times as much, and twenty-five times the Euler load, for the same weight of steel plus a lattice.

Set against a 4% penalty from double lacing, that is not a close contest. The three places it is taken up:

Where the length is very long. A transmission tower leg, a mast, a crane boom, a falsework tower. At slendernesses where a solid section would be hopeless, the lattice buys an order of magnitude and its own penalty is a rounding error.

Where the load is very large. A mill building’s crane column, a heavy industrial frame. Two or four chords carry the axial load between them and the lattice holds them apart, which is cheaper than rolling or fabricating a single section of the required area.

Where the member must be transported and assembled. A lattice is a kit of small pieces; a plate girder column is a single object needing a low loader and a large crane. That is why lattice masts and towers are laced and welded box columns are not.

And one place it is not taken up at all: where the connection cost dominates. A laced column at a 60° angle over 12 m has 69 panels and 138 diagonal ends, which is 138 welds or 276 bolts for a member with two ends. That is the real price of the four per cent, and it is why battens — with an eighth as many pieces and a shear stiffness eighty times worse — persist despite everything on this page.

Where the model stops #

Engesser’s is not the only expression. Haringx’s alternative, $P_{cr} = S_v\left(\sqrt{1 + 4P_E/S_v} - 1\right)/2$, differs in what it assumes about how the shear force follows the deformed member, and it gives a higher answer. The difference matters only when $S_v$ and $P_E$ are comparable, which is exactly the battened case — so the one column where the answer matters most is the one where the two formulations disagree by most.

The chords have their own buckling problem. Between two lacing points a chord is a column of its own, of length $a$ and its own radius of gyration, and it can buckle locally before the whole member does. That check is why lacing panels are short and why it is not enough to make the lattice stiff.

The lacing carries a real force. A built-up column bows, the bow generates a transverse shear, and the lattice carries it — so lacing is designed for a shear that no applied load contains, usually taken as a percentage of the axial force. It is an imperfection-driven force, and it exists because the column is not straight.

And nothing here is inelastic. All of the above uses the elastic modulus, and a stocky built-up column yields before any of it applies.

The shear stiffness is a ceiling, not a correction #

Engesser’s expression has a property worth reading off it before any numbers are put in:

$$\frac{1}{P_{cr}} = \frac{1}{P_E} + \frac{1}{S_v} \quad\Longrightarrow\quad P_{cr} < S_v$$

always, whatever the chords do. Send $P_E$ to infinity — infinitely stiff chords, infinitely far apart — and the critical load approaches $S_v$ and never reaches it. The shear stiffness is not a correction applied to the Euler load; it is a hard upper bound on the whole member.

That reframes what the lattice is for. The chords and their spacing set $P_E$, which is the term everybody looks at and which can be made enormous. The lacing sets $S_v$, which is the term nobody looks at and which caps the answer. A built-up column can be made no stronger than its lattice, and no amount of spreading the chords will get past it.

The sweeps in this essay rise past their own initial $S_v$ only because a battened column’s shear stiffness is not independent of the geometry — the battens get longer as the chords move apart, and their contribution changes with it. For a laced column with the diagonal angle held constant, the panel and the diagonal both scale with the spacing and $S_v$ comes out very nearly constant. There the ceiling is flat, and the curve of capacity against spacing flattens onto it.

There is a check available in that bound and it costs nothing. Any computed critical load above the shear stiffness is arithmetically impossible, so a result that exceeds $S_v$ is an error rather than a good answer — and the error is most likely a factor in the lacing geometry, which is the term with the most opportunities to go wrong.

Which gives the design instruction in one line. Compute $S_v$ first. If it is well above the load, the column is an ordinary column with a correction; if it is anywhere near the load, nothing about the chords will help and the lattice is the design.

The correction is not confined to built-up members #

The same shear flexibility exists in a solid column, and putting a number on it says why nobody mentions it.

For a solid section the shear stiffness is $S_v \approx GA_s$, which for steel is about $0.32EA$. Against the Euler load,

$$\frac{P_E}{S_v} = \frac{\pi^2 EI/L^2}{0.32EA} = \frac{\pi^2}{0.32}\left(\frac{r}{L}\right)^2 = 30.8\left(\frac{r}{L}\right)^2$$

At a slenderness of 100 that is 0.003 — a three-tenths of one per cent reduction, which no calculation would carry. At a slenderness of 10 it is 0.31, a 24 per cent reduction — and a column that stocky fails by squashing long before buckling is relevant.

So the correction is negligible where buckling governs and irrelevant where it does not, which is why Euler’s formula is written without it and why nobody notices its absence.

The built-up column breaks that comfort by being a member whose $S_v$ is three orders of magnitude smaller than a solid section’s while its $P_E$ is as large as ever. It is not a different phenomenon; it is the same term, on a member arranged so that the term matters — which is the general shape of every result in this collection about built-up sections. Separating the material buys the stiffness and creates the flexibility, and the two arrive together because they are two readings of the same arrangement.

The extreme case is worth naming because it is a member nobody calls a column. A lattice mast is a built-up column with four chords and a very large spacing, so its $P_E$ is enormous and its capacity is governed by $S_v$ almost entirely — which is why a mast’s bracing is designed with the same care as its legs, and why a mast with a diagonal missing is in far more trouble than the missing member’s own force would suggest. The diagonal was not carrying much; it was supplying part of a shear stiffness that is the whole of the answer.

The same arithmetic, three places on this site #

The harmonic sum is worth recognising as a shape rather than as a formula, because it turns up wherever two mechanisms act in series along one load path.

Rankine’s column formula, $1/P = 1/P_y + 1/P_{cr}$, combines squashing and buckling the same way, and it was the practical column formula for the best part of a century for exactly the reason Engesser’s works: the smaller capacity governs smoothly and the transition is automatic.

A beam and its supports, where the two flexibilities add and the softer one decides how much of the movement belongs to which.

And a composite beam’s two limits, where a partial shear connection puts the interface stiffness in series with the section’s own.

In all four the useful reading is the same and it is about what to improve. A series combination is dominated by its weakest term, so effort spent on the stronger term is largely wasted — and the arithmetic says how largely. On the battened column above, doubling the chord spacing to 600 mm quadruples the Euler load, from 1,341 to 5,228 kN, and takes the critical load only from 312 to 661: a factor of four in the strong term bought a factor of 2.1 overall. Halving the batten spacing instead multiplies the shear stiffness by 7.5, from 406 to 3,027 kN, and takes the critical load to 930 — a factor of 3.0, from the term that was governing.

The question a series combination asks is never “how much” but “which one”.

What the pictures cannot show #

The two modes in the hero figure are drawn separately, and they never occur separately. The real deformed shape is a single curve containing both, in a proportion set by the ratio of the two stiffnesses, and drawing that would show one curve where the argument needs two.

The lattice figures draw a panel count chosen for legibility. A real laced column at a 60° angle and 300 mm spacing has a panel of 173 mm over a 12 m length, which is 69 panels — too many to draw and exactly the number that makes the shear stiffness what it is.

And the sweeps run the chord spacing to three metres, at which point the object is not a column any more but a small tower with two legs. The curve is continuous and the engineering is not.

The ladder from here #

Later rungs on this anchor: the local buckling of a chord between lacing points, and the interaction between it and the overall mode. Haringx against Engesser, and the elastomeric bearing where the disagreement is largest. The shear force in the lacing, and why it is specified as a fraction of the axial load rather than computed. Four-chord towers and masts, where the same arithmetic is applied about two axes at once and the lacing planes interact. Built-up columns in timber, where the connection is nails and the slip modulus does the work of the lacing area. And the historical case: laced and battened columns were the standard form for a century of iron and steel construction, and Engesser’s correction was published in 1891 after built-up columns kept failing at loads their second moment of area said were safe.