Structural Engineering & Statics
where the load goes
Sections and stress

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

Four sections of identical area, identical weight and identical cost. The stiffest is dozens of times the stiffest of the flattest, and the only thing that changed was the arrangement.
10 min read 7 figures Geometry beats materialWhich failure arrives first

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

The same material, four waysFour cross-sections of identical area, so identical weight and cost, with the second moment of area computed from each profile's own geometry. Only the arrangement differs, and the stiffest is many times the flattest.the same, laid flatI = 0.06 × 10⁶1.0× the firstsquareI = 0.75 × 10⁶13.3× the firsttall rectangleI = 10.00 × 10⁶177.8× the firstI-sectionI = 24.29 × 10⁶431.8× the firstevery section here has an area of 3000 — only the shape differsthe bar is the second moment of area, to scale
Fig. 1 Four cross-sections of identical area, so identical weight and identical material cost, with the second moment of area computed from each profile’s own geometry. Only the arrangement differs.

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, ∫y2 dA\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.

Every strip counts by the square of its distanceA rectangular section divided into equal strips, with each strip's contribution to the second moment of area drawn beside it. The strips are identical in size; only their distance from the neutral axis differs.neutral axiscontribution of each striptotal I = 79.86 × 10⁶the outer strips do almost all of the work
Fig. 2 The reason for the ranking, in one picture. Each strip contributes in proportion to the square of its distance from the neutral axis, so a section is judged by how much of its area it can get far away.

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.

Moving the flanges apartThe second moment of area of an I-section against its depth, with the flange and web areas held constant. The growth is close to quadratic, because the parallel-axis term dominates everything the flanges contribute about their own centres.1001502002503000M10M20M30M40M50M60Moverall depth1.0×2.6×4.9×7.9×13.8×21.3×same steel, moved apart
Fig. 3 Second moment of area against depth for an I-section with the flange and web areas held constant. The curve keeps rising and the material does not change — but somewhere along it the section stops being able to hold its own shape.

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.

The column curveFailure load against slenderness, as a fraction of the squash load. A stocky column crushes; a slender one buckles at the Euler load; the crossover is where the two curves meet, and real columns fall below both near it.5010015020000.20.40.60.811.2slenderness (effective length ÷ radius of gyration)they cross at λ = 75squashingEuler bucklingreal columns, which are neither
Fig. 4 Failure load against slenderness. A column is judged by its weakest axis, which is why the shape that is best in bending is not the shape that is best in compression.

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.

Bending is a push and a pullA section carrying a bending moment, with the stress at every height computed as the moment times the distance from the neutral axis divided by the second moment of area. It is compression above and tension below, and zero exactly at the neutral axis.neutral axiscompressiontensionI = 29.97 × 10⁶Z = 299.7 × 10³peak stress 200.2σ = M y ÷ I, at every height
Fig. 5 The elastic stress block in an I-section. Almost all the material is near the peak stress already, which is what makes the section efficient and what leaves it little to recruit after first yield.

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.

The shape has to survive being made

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

Length costs more than it looksThe same column section at four lengths, with the buckling capacity of each drawn as a bar. Capacity falls as the inverse square of the length, so a column three times as long carries a ninth as much.1× the length100% of the capacity1.5× the length44% of the capacity2× the length25% of the capacity3× the length11% of the capacityidentical section, identical material, identical end conditions
Fig. 6 Capacity against length for a compression member. Local buckling is the same phenomenon at the scale of a single plate, and it arrives sooner the thinner the plate is.

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 deflected shape is the moment, integrated twiceA loaded beam with its deflected shape above and its bending-moment diagram below. The shape was obtained by integrating the moment twice and fitting the constants to the supports; the vertical scale is exaggerated enormously.the largest movement, at x = 4.00momentdrawn at roughly three hundred times the real deflection —a beam at its serviceability limit moves about a three-hundredth of its span
Fig. 7 A deflected beam. The section that resists this has to hold its own shape while doing so, which is what the width-to-thickness limits are protecting.

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.

Where this goes next

neutral axiscontribution of each striptotal I = 79.86 × 10⁶the outer strips do almost all of the work The material far from the middle does nearly all the work A strip of steel contributes to bending stiffness in proportion to the square of its distance from the centre. Move the same steel outward and the section gets stiffer for nothing. neutral axiscompressiontensionI = 29.97 × 10⁶Z = 299.7 × 10³peak stress 200.2σ = M y ÷ I, at every height Bending is a pair of forces, pushing and pulling A bending moment is not a mysterious twisting. It is a push near the top of a section and a pull near the bottom, separated by a lever arm — a couple, made out of stress. 5010015020000.20.40.60.811.2slenderness (effective length ÷ radius of gyration)they cross at λ = 75squashingEuler bucklingreal columns, which are neither Strong enough and still falls over A column can fail at a fraction of the load its material could carry, by going sideways. Buckling is a failure of stability rather than of strength, and it is decided by geometry.