The one length a section takes into a column

A cross-section is a rich object. It has an area, a centroid, a second moment about every axis through that centroid, a product of inertia, two section moduli, a plastic modulus, a shear centre, a torsion constant, a warping constant. Bending uses several of them. Shear uses another. Torsion uses two more.

A column uses one, and it is not on that list. It is $r = \sqrt{I/A}$, a length, and every other property of the section reaches a column only through it.

Why only one property gets through #

The Euler load is $P_{cr} = \pi^2 EI / L^2$, which contains $I$ and not $r$, so it is fair to ask where the radius of gyration comes in at all.

It comes in the moment the question becomes which failure arrives first. A column has two limits: it crushes at $P_y = A f_y$, and it buckles at $\pi^2 EI/L^2$. Divide both by the area to turn them into stresses:

$$\sigma_{squash} = f_y, \qquad \sigma_{cr} = \frac{\pi^2 E I}{A L^2} = \frac{\pi^2 E}{(L/r)^2}$$

and the section has vanished. What is left is $E$, which is a material, and $L/r$, which is a single number called the slenderness. Two columns with the same slenderness fail at the same stress whatever their shapes, whatever their sizes, whatever their areas.

That reduction is worth pausing on because it is unusually complete. A beam needs $I$ for deflection, $Z$ for strength, $Z$ again for the other face if the section is asymmetric, the shear area for the shear check, and the shear centre if it might twist. A column needs a length.

What the length is #

$r = \sqrt{I/A}$ has an interpretation that makes it far less arbitrary than the formula suggests.

Take all the material in the section and imagine it condensed into two equal lumps, symmetrically placed either side of the centroid. Put them at $\pm d$. Their second moment is $A d^2$. Ask what $d$ gives the same second moment as the real section, and

$$A d^2 = I \implies d = \sqrt{I/A} = r$$

So the radius of gyration is the distance at which the whole area would have to sit to be as stiff as the real section is. It is a summary of how far away the material is, measured in the only way a second moment cares about — root mean square distance from the axis.

The construction underneath it is the ordinary one: each strip’s area times the square of its distance from the axis, summed. What the division by $A$ adds is that the sum becomes a root-mean-square of those distances rather than a total of them — a single length standing for a whole distribution, and one that does not grow when the section does. And because moving material outward raises $I$ as the square of the distance while leaving the area alone, it raises $r$ linearly: making a section deeper is exactly making its column better, at a rate of one for one.

For simple shapes it comes out as a fixed fraction of a dimension, and the fractions are worth having by heart:

Every one of them is a fraction of a dimension of the section, and none is far from a quarter. That is why designers can estimate slenderness from a drawing: an I-section 300 mm deep has $r_x$ near 125 mm, and a 6 m length of it is at a slenderness of about 48 without opening a catalogue.

The section that is worse as a beam is better as a column #

The tee and the rectangle in the hero figure have the same area, 3,000 mm², and the same depth, 200 mm.

As a beam the tee is a quarter weaker, because its centroid has moved toward the flange and the far fibre is now further away. As a column it is six per cent better, because the same movement of material raised the second moment and the area did not change.

The reason the two rankings differ is exactly one division. Strength is $I/c_{\max}$ and stiffness-in-compression is $I/A$; concentrating material near one face raises $I$, raises $c_{\max}$, and leaves $A$ alone. So it helps one ratio and hurts the other.

Bending is a push and a pull on a lever arm, and the lever arm reaches to the furthest fibre; compression has neither. A column has no extreme fibre. Nothing about compression buckling asks how far the furthest material is from the axis, only how far the average material is — and that is the whole of the difference.

Put the same two on the beam ranking and the order reverses, without a single number about either section changing.

The range, and what it costs #

Across the five shapes of equal area, $r$ runs from 4.33 mm to 89.98 mm — a factor of 20.8. As a 4 m pin-ended column the Euler loads therefore run from 7 kN to 3,146 kN, a factor of 432, because the load goes as $r^2$.

Nothing about the amount of material changed anywhere in that range. It is the same 3,000 mm² of steel, and the difference between the worst arrangement and the best is nearly three orders of magnitude of load — which is the geometry-beats-material argument at its most extreme anywhere in this collection.

The length costs in the same currency, and the arithmetic is the same arithmetic. Capacity goes as the inverse square of the length exactly as it goes as the square of the radius, so a column three times as long, or with a third of the radius of gyration, carries a ninth as much. Doubling the length is therefore indistinguishable from halving $r$, and the picture says so.

The same length generates the kern #

The radius of gyration is not confined to columns, and the place it turns up next is one this collection has already visited under a different name.

The kern is the region within which the resultant of a compression must fall if no part of the section is to be pulled, and its boundary sits at $Z/A$ from the centroid in each direction. Substituting $Z = I/c$:

$$k = \frac{Z}{A} = \frac{I}{Ac} = \frac{r^2}{c}$$

The kern distance is the radius of gyration squared, divided by the distance to the opposite extreme fibre. For a rectangle, $r^2 = h^2/12$ and $c = h/2$, so $k = h/6$ — the middle third, arrived at from a column property. For a thin ring, $r^2 = D^2/8$ and $c = D/2$, so $k = D/4$, and the generous kern of a chimney is a consequence of its generous radius of gyration.

The relation is worth having for what it says about the two quantities. They are the same information read for two purposes: $r$ measures how far the material is from the axis in a root-mean-square sense, and $c$ measures how far the furthest material is. Their ratio decides how much eccentricity a no-tension section tolerates, and their absolute values decide how a column behaves — so a shape that is good as a column is, by the same arithmetic, a shape with a generous kern.

It also connects the two subjects on one member. A column carrying its load at an eccentricity $e$ has no tension anywhere provided $e < r^2/c$ — and a slender column bends, which adds its own deflection to $e$. So a column that starts inside its kern can leave it as it deflects, and the section that was wholly in compression at the ends is not at mid-height. For a material that can be pulled this is a stress question; for masonry it is a contact question, and the same $r^2/c$ decides both.

Which explains a pairing that looks like a coincidence in the catalogue. A circular hollow section is the best common shape as a column and the best common shape for an eccentrically loaded masonry-like member, and it is both for one reason: it has the largest $r$ for its outline and its $c$ is no larger than anything else’s.

The axis every code plots against #

There is a second reduction on top of the first, and it is the one that lets a single curve serve every material.

The two limits cross where the squash stress equals the Euler stress:

$$f_y = \frac{\pi^2E}{(L/r)^2} \quad\Longrightarrow\quad \lambda_1 = \pi\sqrt{\frac{E}{f_y}}$$

which is 93.9 for S235 steel, 76.4 for S355, and 67.1 for S460. It contains no section: it is a property of the material alone, and it is the slenderness at which the shape stops deciding and the strength starts.

So the two variables separate cleanly. The section enters through $r$ and nothing else; the material enters through $\lambda_1$ and nothing else; and the ratio

$$\bar\lambda = \frac{L_e/r}{\lambda_1} = \sqrt{\frac{A f_y}{N_{cr}}}$$

is dimensionless in both — the second form being the one codes write, and being the square root of the ratio of the two capacities the first form compares. Every modern column curve is plotted against that quantity, with 1.0 at the crossover, and one picture then serves every steel and every profile.

Two things follow that are worth stating plainly. The ratio, not the slenderness, is what a designer should carry. A slenderness of 80 is stocky in mild steel and slender in a high-strength grade, and quoting it without the grade says nothing. And the normalisation is why the design curves look like curves rather than a corner: with the two straight-line limits scaled to meet at $\bar\lambda = 1$, everything the real data does — the imperfections, the residual stresses, the rounding of the knee — shows up as a single family of curves in the neighbourhood of one, which is the only region a test programme needs to cover.

It is a small piece of bookkeeping and it is the reason a column can be designed from a table rather than from an argument.

The smaller radius is the only one that matters #

A section has a radius of gyration about every axis through its centroid, and it has a smallest one.

An I-section chosen for bending is a poor column and the arithmetic says how poor. Its second moment about the minor axis is a small fraction of the major one, so $r_y$ is a small fraction of $r_x$, and a member with a slenderness of 45 about the axis it was chosen for has 100 or more about the axis nobody looked at. That is how a member strong enough for its load falls over anyway.

The consequences are entirely practical. A column section is a different shape from a beam section — square hollow sections, circular hollow sections, and universal columns whose flanges are as wide as they are deep, rather than universal beams. And a beam pressed into service as a column needs restraint about its minor axis, which is the brace that need not be strong doing its familiar job of halving a length rather than adding a strength.

Which puts the other factor of the same ratio into view. Slenderness is $L_e/r$, and the effective length depends on what holds the ends, so improving a column is a choice between a better section, a shorter length and a better restraint — three routes to one number, exchangeable at a fixed rate, and the cheapest of them is usually not the section.

A third radius, for a mode that is not bending #

The reduction to a single number is not quite complete, and the exception is worth stating because it is the one case where a section’s other properties come back.

A channel has three critical loads rather than one. Two are flexural, about the two principal axes, and use the two radii of gyration this page is about; the third is torsional, about the shear centre, and uses the torsion constant, the warping constant and the polar radius of gyration about that centre. The lowest of the three governs, and on a channel of ordinary proportions it is the twist.

For a section whose shear centre is not at its centroid — a channel, an angle, a tee, any singly symmetric shape — the flexural and torsional modes couple, and the governing critical load is below the lowest of the three uncoupled ones. The quantity that decides it is the polar radius of gyration about the shear centre, which is a fourth length and involves the distance between the two centres.

So the honest statement is: a doubly symmetric section takes one length into a column; anything else takes three or four. The single-number reduction holds for the sections most columns are made of, and fails for exactly the sections that are not.

Making the radius larger without adding material #

The last consequence is the one that produced a whole family of members.

Since $r$ rewards distance and $A$ does not, a section can be improved as a column by moving its material apart — and if the material is moved apart far enough, the pieces stop being one section and become two chords joined by something.

Where that ends is a column built out of two columns, whose radius of gyration is essentially half the spacing of its chords and can be made as large as the connections allow. The Euler load climbs with the square of the spacing exactly as $r^2$ says it should — and then runs into a ceiling that has nothing to do with the section at all, because the lacing that holds the chords apart has a stiffness of its own and it is that stiffness the member eventually fails on.

That limit is the proper end of this essay’s argument. The radius of gyration is the whole of what a column knows about its section, so the way to make a better column is to make $r$ bigger; and pushed to its limit, that instruction stops being about sections and becomes about lattices.

Where the quantity came from #

The radius of gyration is borrowed, and the borrowing shows in the name.

It arrives from dynamics, where a rigid body rotating about an axis has a moment of inertia $I_m = \sum m r^2$ and the radius of gyration is the distance at which the whole mass would give the same moment of inertia. The second moment of area is the same sum with area in place of mass, and it appears in bending for a completely different reason — it is the integral of $y^2\,dA$ that turns a linear stress distribution into a couple, and nothing in that derivation is rotating.

Two quantities with the same algebra and no shared physics, sharing a name and a symbol. This collection has met the pattern before, and it is worth naming again: the word “inertia” in “moment of inertia of a section” is a fossil, and a reader who takes it literally spends some time looking for the motion.

What the borrowing did supply is the summary idea. A distribution of area about an axis is a complicated object; the second moment collapses it to one number, and dividing by the area collapses it to one length, which can be compared with the member’s own length. That comparison is the whole of column design, and it was not available until somebody noticed that the ratio of two lengths was the governing variable.

The intervening history is instructive about what a good variable is worth. Euler published the buckling load in 1744 and it was widely disbelieved, because it predicts an infinite capacity for a short column and real short columns crush. Rankine and Gordon’s formula of the 1860s, which combines the two limits into $1/P = 1/P_y + 1/P_{cr}$, was the practical answer for the best part of a century — and it works precisely because it is a function of $L/r$ alone. Every empirical column formula of that period is a curve on the same axis, and the axis is this quantity.

Note the shape of that combination, incidentally: two capacities in series, the smaller governing, exactly as the built-up column’s two stiffnesses add as flexibilities rather than as strengths. It is the same arithmetic doing the same job in two places a century apart.

Where the model stops #

$r$ is elastic. The reduction to $L/r$ comes from the Euler load, which assumes the material is still on its elastic modulus. A stocky column yields first, at which point the governing stiffness is a tangent modulus and the column has already partly failed before it was loaded.

The area in $\sqrt{I/A}$ is the gross area. A section with holes has a smaller area and, usually, a smaller second moment, and the two do not shrink in the same proportion — so the net section that decides the tension capacity is not the section that decides the slenderness.

And a slenderness is only meaningful with an effective length beside it. $L/r$ where $L$ is the member’s own length is a number about a pin-ended column and nothing else; every real column needs the $L_e$ its restraints deserve, and getting that wrong moves the answer far more than any choice of section does.

What the pictures cannot show #

The hero figure draws $r$ as a pair of dashed lines either side of the centroid, which is the honest geometric reading and looks like an edge of something. There is no material at that distance in any of the five sections; it is where the material would have to be.

Nor can any of these figures show that the ranking they draw is for one axis of each section. The I-section, which is best by a wide margin in every picture here, is the worst of the five about its own minor axis — and that is the axis its column would buckle about.

The ladder from here #

Later rungs on this anchor: the polar radius of gyration and the torsional modes that need it. The effective radius of a built-up section, and why a code adds a term to the slenderness of a laced column rather than letting the geometry speak. The slenderness limits that appear in design codes as absolute numbers — 180, 250 — which are not calculations at all but conventions about handling, transport and vibration. Local slenderness, $b/t$, which is the same idea applied to a plate rather than a member and decides whether the section can reach its own strength. And the historical case: Euler published in 1744, Lamarle showed in 1845 which of the two limits applies where, and the intervening century of empirical column formulae was spent inventing the quantity this essay is about.