The same steel in a different shape, and a factor of forty

A tonne of steel costs the same whatever shape it is rolled into. What it can carry does not.

The four profiles above enclose exactly the same area. Their bending stiffnesses differ by a large factor, and the difference is free — no better steel, no more of it, no cleverer connection. It is the single most reliable lever in structural design, and the reason a steel catalogue contains hundreds of shapes rather than a range of solid bars.

The ranking, and why it is that order #

Flat. The worst possible arrangement for bending about the horizontal axis: all the material close to the neutral axis, contributing almost nothing. Its only virtue is that it is very good about the other axis, which is why a flat plate is the right shape when the bending is in its own plane.

Square. Better, because half of it is now some distance out. Still poor, because the central material is close to useless and it is a large fraction of the total.

Tall rectangle. Better again, and by the cube of the depth. This is the ruler on edge, and it is the whole of the timber joist’s design logic.

I-section. The end of the argument. The material is at the extremes where the stress is highest and the lever arm longest, and the web is reduced to what is needed to hold the flanges apart and carry the shear.

The ordering is not a matter of taste. It is the second moment of area, $\int y^2\,dA$, evaluated for each, and the numbers under the drawings are that integral computed from the rectangles that make up each profile.

Why not thinner still #

If moving material outward is free, the obvious question is why the flanges are not paper-thin and a metre apart.

Four things stop it, and each is a different failure mode arriving first.

Local buckling. A thin plate in compression buckles as a plate, independently of whether the member as a whole is stable. A flange too wide for its thickness ripples; a web too deep for its thickness folds. Section classification in every steel code is a set of width-to-thickness limits, and it exists precisely to stop a designer taking this argument to its conclusion.

Shear. The web carries nearly all the shear, and a web thinned to nothing has nothing to carry it with. Near supports, where the moment is small and the shear is large, the web governs.

Lateral-torsional buckling. A deep, narrow beam in bending can buckle sideways and twist, failing at a moment well below its section capacity. The remedy is restraint along the compression flange, and the need for it grows exactly as the section is made more efficient.

Making it. Rolling has practical limits on how thin a web can be while a section is being formed hot, and welding thin plate distorts.

That curve is the argument for depth and, read carefully, also its limit. It grows without bound because the model has no idea that the web is getting thinner as the depth grows at constant area. Real sections leave the curve well before its right-hand end.

Hollow sections, and the other axis #

The I-section is optimal for bending about one axis and poor about the other, which is fine for a beam and bad for a column.

A column is loaded axially and can buckle in any direction, so it is governed by its weakest axis. An I-section has a strong axis and a weak one differing by a factor of three or more, and as a column it can only use the weak one — the strong axis’s advantage is wasted.

The answer is a shape with no weak axis: a circular or square hollow section. Its material is uniformly far from the centre in every direction, so it has the same second moment about every axis, and as a column it is far more efficient than an I-section of the same weight.

That is why buildings are framed with I-sections as beams and hollow sections or H-columns as columns, and why the two families exist. The governing failure is different, so the optimal shape is different.

What the shape does after yield #

Shape also decides how much reserve a section has past first yield.

The shape factor is the ratio of plastic to elastic moment capacity, and it measures how much under-used material sits near the neutral axis waiting to be recruited. A rectangle has 1.5 — half as much again after the extreme fibre yields. An I-section has about 1.14, because it had little spare material to begin with.

So the efficient shape is efficient at the elastic stage and has less in reserve, which is a genuine trade rather than a free lunch. A structure designed plastically gets less from an I-section than the elastic comparison suggests.

So the ranking reverses. The shapes that win the elastic comparison this essay opened with are the ones with the least left over afterwards, which is exactly backwards from the way the plastic reserve is usually described — as a bonus that good sections have more of.

The four numbers, and the two different ratios #

The figure at the top holds the area at $3{,}000\,\text{mm}^2$ in every case and computes each profile’s properties from its own rectangles. The results are worth reading rather than glancing at, because they contain a distinction the ranking hides.

The stiffness ratio from worst to best is 432. The strength ratio, comparing section moduli, is 32. The same four pieces of steel, ranked by two properties, differing by more than a factor of thirteen in how much the ranking spreads them.

The reason is the division by $c$. Strength is $I/c$, and the shapes that have a large $I$ have it partly because they are deep, which makes $c$ large as well and gives some of the gain back. Stiffness keeps the whole of the depth advantage; strength keeps only the part of it that is not spent on being far from the neutral axis.

Two consequences follow. First, shaping a section is worth more for a deflection-governed member than a strength-governed one, which reinforces the observation that long spans are governed by movement — the lever that helps is the one that acts on the limit that governs. Second, comparing two sections on the wrong property is an easy mistake with a large answer: a section chosen for its $I$ when the member is strength-critical, or the reverse, can be a size out.

The flat profile is worth one more look. Laid flat it is the worst thing in the table by a factor of hundreds; stood on edge it is the tall rectangle, the second best. One piece of steel, two entries in the table, differing by 178 times in stiffness, and the difference is a rotation of ninety degrees. The catalogue of sections is, in a sense, a catalogue of decisions about which way up the material sits.

The plastic reserve, on the same cut face #

The shape factor was quoted above as a ratio. Deriving it takes one free body and a change of assumption, and the derivation shows where the reserve is hiding.

Cut the beam and look at the exposed face. In the elastic state the stress varies linearly from zero at the neutral axis to $\sigma_y$ at the extreme fibre, and only the outermost sliver of material is at its limit. Now go on loading. The outer fibres cannot take more stress, so they hold at $\sigma_y$ while the strain grows, and the yielded region spreads inward. In the limit the whole of the compression side is at $\sigma_y$ and the whole of the tension side at $\sigma_y$: two rectangular blocks rather than two triangles.

For a rectangle $b$ wide and $d$ deep, that limit is a couple. Each block has a force $\sigma_y b d/2$, and their centroids are $d/2$ apart, so

$$M_p = \sigma_y\,\frac{bd}{2}\cdot\frac{d}{2} = \frac{\sigma_y b d^2}{4},$$

against the elastic $M_e = \sigma_y bd^2/6$. The ratio is $6/4 = 1.5$, and it came from nothing but the two stress blocks having different centroids — a triangle’s centroid sits at two-thirds of the half-depth, a rectangle’s at one-half.

That is why the shape factor is a measure of idle material. A rectangle has a great deal of steel sitting near the neutral axis at low stress, and the plastic state recruits all of it. An I-section has almost none, so its factor is around 1.14. A tee is the extreme case in the other direction, with a factor that can exceed 1.8, because its stem is nearly all near the axis and nearly all wasted elastically.

The trade is exact and worth stating in one line: the efficiency of a section and its reserve after yield are the same quantity, read in opposite directions. Material near the neutral axis is wasted elastically and available plastically. There is no shape that is good at both, and choosing between them is choosing whether the design is governed by first yield or by collapse.

The catalogue is discrete #

Everything above treats the section as a continuous variable. It is not; it is a list, and the list has a structure of its own that shapes what actually gets built.

Rolled sections come in serial sizes, and within a serial size the rolls are the same. Extra weight is added by opening the rolls a little, which thickens the flanges and the web and pushes the overall depth out while the inner profile between the flanges stays constant. That is the reason a family of sections nominally 457 mm deep runs from about 449 to 465 mm: the depth is an output of the rolling, not a target. It is also a considerable convenience, because a connection detail, a notch or a fitting designed against the inner profile works for every member in the family — which is why fabricators think in serial sizes and analysis programs think in section properties.

Two practical consequences follow from the list being discrete. The first is the “next size up” penalty: a member 4 per cent overstressed cannot be made 4 per cent bigger, and the available step is often 10 to 15 per cent of weight. A great deal of structural steel in existence is a size larger than the calculation required, because nothing between the two exists.

The second is that the right selection criterion is weight per metre rather than any structural property. Steel is bought by the tonne, so the cheapest adequate member is the lightest adequate member — and the lightest adequate member is very often not the one with the smallest adequate section property, because the families are not ordered consistently across properties. Selection tables are therefore printed sorted by mass, with the section properties beside them, which looks like a strange way to organise a structural handbook until the reason is visible.

Two things follow that a continuous treatment cannot see. The sensitivity of a design to an assumption is nearly zero over most of a tread and enormous at a riser, so the same one per cent of load matters not at all in one member and buys a whole size in the next. And an optimisation that returns three significant figures is answering a question with twelve answers in it.

The same argument outside steel #

Nature reached the same conclusions.

A bone is a hollow tube with marrow in the middle, which is the hollow-section solution to a member that bends in unpredictable directions. A bamboo culm is a thin-walled tube with nodes at intervals — the nodes being the answer to local buckling, which is exactly the failure mode a thin tube is prone to. A blade of grass, a wheat stalk and a bird’s wing bone are all thin-walled tubes with internal stiffening.

A corrugated sheet is the same depth argument applied to a plate: folding it moves material away from the mid-plane and multiplies the stiffness about one axis by a large factor, at no cost in material. Cardboard, roofing sheet and aircraft skin are all the flat section from the first figure, folded until it is one of the better ones.

One number for a shape, with the size divided out #

The comparison above holds the area constant so that the shapes can be ranked. There is a way to remove the size entirely and get a single dimensionless number for a shape, and it is worth having because it makes the ranking portable.

$I$ has dimensions of length to the fourth and $A^2$ has the same, so their quotient is a pure number. Normalise it so that a solid circle scores one:

$$\phi = \frac{4\pi I}{A^2}$$

A solid circle gives exactly 1 by construction. A square gives $4\pi/12 = 1.05$ — a shape is barely improved by having corners. And a thin-walled tube of radius $r$ and wall $t$ gives $I = \pi r^3 t$, $A = 2\pi rt$, and

$$\phi = \frac{4\pi\cdot\pi r^3 t}{4\pi^2r^2t^2} = \frac{r}{t}$$

The efficiency of a tube is its radius-to-thickness ratio, exactly, with nothing else in it. A tube with a wall a fortieth of its radius is forty times as efficient in bending as the same steel drawn into a rod, and the number is legible off a drawing without any calculation at all.

Rolled I-sections score between about 10 and 25 on the same measure, and a fabricated plate girder can reach 50. Which is why the ceiling matters: $\phi$ is bounded by local buckling, and for structural steel the practical maximum is somewhere in the forties to sixties. The shape factor and the plate slenderness are the same variable read two ways — pushing one up is pushing the other up, and the section catalogue is the record of where the profession stopped.

The shape has to survive being made #

Every argument above assumes the section keeps its shape while it works, and thin plates do not.

A flange too wide for its thickness buckles as a plate long before the member buckles as a column, and the capacity that the second moment of area promised is never reached. Slenderness governs at both scales, and section classification is a set of limits written to keep the small-scale version out of the calculation.

The consequence is that the curve of stiffness against depth is not followed to its end, and the practical span-to-depth ratios reflect where the plates stop cooperating rather than where the arithmetic stops improving.

Where the model stops #

Equal area, not equal cost. Rolling an I-section costs more per tonne than rolling a bar, and fabricating a plate girder costs a great deal more. The material is equal; the price is not.

Bending about one axis. The comparison ranks the profiles about the horizontal axis only. About the vertical axis the ranking is nearly reversed, and any member that might be loaded either way has to be judged on both.

No local buckling in the arithmetic. The second moment of area is computed as though every plate stays flat. Section classification exists because they do not, and a class 4 section cannot reach the capacity its geometry implies.

Elastic only. As above, and the stress block is assumed triangular throughout.

No shear, no torsion. An open section such as an I-beam is very poor in torsion — hundreds of times worse than a closed tube of the same area — and nothing in the bending comparison shows it.

The figures have a limitation worth naming explicitly: the four profiles are drawn at the same scale and look like reasonable members, but the equal-area constraint makes the I-section’s web about two millimetres thick at the depth shown. No such section is rolled. Holding the area constant is the right way to make the comparison and it produces a profile nobody would build, which is a good reminder that a fair comparison and a realistic one are not the same.

The ladder from here #

Later rungs: section classification and local buckling limits. Effective width, and how a buckled plate is accounted for. Shear in webs, and web stiffeners. Torsion in open and closed sections. Hollow sections and the columns that use them. Shape factors and plastic design. Castellated and cellular beams. Composite sections in steel and concrete. And optimisation proper, where the section is generated rather than selected — which turns out to converge on shapes that look like bone.

The first wrought-iron I-beams were rolled in 1849 and the shape was contentious, because the material away from the flanges was thought to be doing something. The catalogue of standard sections that followed did more for structural efficiency than any improvement in the iron itself.