The ranking belongs to the load case

Steel has a modulus of 210 GPa and timber about 11. That is a factor of nineteen, and it is the number every comparison of the two materials starts from.

For a beam of given stiffness and given span, free to choose its own depth, timber wins by 4.28 to one.

Nothing about either material has been misquoted. What has happened is that the question was asked properly.

Which free body produced the number #

Not a free body this time — a derivation in three lines, and it is the same three lines every time.

Write the mass. Write the constraint. Eliminate the free variable. Read off what is left.

For a beam of length $L$, of given stiffness $S$, free to choose its cross-sectional area $A$:

$$m = \rho A L, \qquad S = \frac{C E I}{L^3}$$

For a section of fixed shape, $I \propto A^2$ — a square of area $A$ has $I = A^2/12$. So $A \propto \sqrt{S L^3 / CE}$ and

$$m \propto \rho L \sqrt{\frac{SL^3}{CE}} \;\propto\; \frac{\rho}{E^{1/2}}$$

Minimising the mass means maximising $E^{1/2}/\rho$, and that is the index. The exponent is not a fit and not a rule of thumb: it is exactly one half because $I$ goes exactly as $A^2$ for a shape that is scaled rather than changed.

Change what is free and the exponent changes with it:

and the strength versions replace $E$ by $\sigma$ with exponents 1, $2/3$ and $1/2$.

Why the exponent decides everything #

A tie has no shape to choose. Its area is whatever the load and the stress dictate, so a material’s mass is proportional to $\rho/E$ directly and steel’s nineteen-fold modulus advantage is answered by its nineteen-fold density: steel and timber tie almost exactly, 1.000 to 0.979.

A beam has a depth to choose. A low modulus can be answered by making the member deeper, and depth is very cheap in a beam — $I$ goes as the square of the area. So the penalty for a low modulus is only a square root of it, while the reward for a low density is undiminished, and timber’s density advantage of nineteen against steel’s modulus advantage of nineteen-under-a-square-root gives $19/\sqrt{19} = 4.36$.

A plate has a thickness to choose, and thickness is cheaper still — $D$ goes as $t^3$. The exponent drops to a third and timber wins by 6.99.

The whole of the ranking is in which power the geometry can be traded at, and the material data is the same in all three columns.

The chart, which makes the construction visible #

Plot $\log E$ against $\log \rho$. An index $E^{1/n}/\rho$ = constant is the line

$$\log E = n \log \rho + \text{const}$$

a straight line of slope $n$. Better materials sit above and to the left of it, and ranking a set means sliding the line until only one is left.

That is Ashby’s construction and it is worth having for a reason beyond the arithmetic: it makes the sensitivity visible. Two materials close to the same guide line are nearly interchangeable on that index however far apart they are in either property, and two materials far apart along a line are not distinguished by it at all. The chart shows which distinctions in a property table matter and which are noise, and no table can.

It also shows why steel and high-strength steel are indistinguishable on every stiffness index — the same $E$, the same $\rho$, the same point on the chart. A stronger steel does not change the one number any stiffness index contains.

The reordering, which is the actual result #

Run all six indices over eight materials and count how far each moves.

Five of eight move four places or more. Glass is second for a tie of given stiffness and seventh for a tie of given strength, because it is stiff and weak — an entirely ordinary combination that no single ranking can represent.

Only carbon fibre stays put, and only because it is better than everything else on every index in the table, which is a fact about it having no weaknesses rather than about rankings being stable.

Where the depth actually goes, which is the part the index hides #

The stiff-beam index says a timber beam of a given stiffness weighs a quarter of a steel one. It does not say how big it is, and the answer matters.

Eliminating $A$ gave the mass. Solving for $A$ instead gives the size, and it scales as $\sqrt{S L^3/E}$ — so the timber beam has $\sqrt{19} = 4.4$ times the cross-sectional area of the steel one and about twice the depth. That is where the win comes from: the material is spread further out, the second moment recovers, and the mass falls because timber is nineteen times lighter and only four times bigger.

Two practical consequences follow, and both are why the index alone does not decide anything.

The floor zone. A timber beam twice as deep as a steel one is a storey height, over enough floors, and a storey height is a building. The index’s free variable is the one a building most often refuses to let go.

The shape factor. The derivation assumed $I \propto A^2$, which is true for a scaled section and false for a changed one. Steel can be rolled into an I-section with $I$ four or five times a square of the same area; timber, in ordinary sizes, cannot. So the same material in a different shape recovers part of steel’s loss, and a fair comparison has to say which shapes each material can be had in.

Ashby’s own treatment handles this with a shape factor multiplying the index, and it is the single largest correction to any of the numbers in this essay: an I-section’s shape factor of four multiplies steel’s stiff-beam index by two, halving timber’s advantage from 4.28 to about 2.1.

What happens when cost is in it #

The indices above minimise mass, which is what an aerospace structure wants and what a building emphatically does not — self weight matters most where it is being carried a long way, and a floor plate is not that. Divide each index by the material’s cost per kilogram and the table inverts almost completely.

Carbon fibre, which was second by mass, is last by a factor of ten. Concrete, which was fifth, is first.

That is not a criticism of the mass indices; it is the same method applied to a different objective, and the whole point of the method is that the objective is stated. A material ranking is a function of what is being minimised and what is being held constant, and a table that names neither is answering an unspecified question.

It also says something honest about why buildings are made of concrete and timber and not of anything better. They are not better. They are cheap per unit of the thing the building actually needs, which is stiffness at a span, and the exponent in that phrase is doing as much work as the price.

The one index everybody uses without naming it #

There is a seventh index in common use that the table above leaves out, and it is worth adding because it is the one an engineer applies most often without noticing.

Span-to-depth ratio. A designer who says “steel spans 20 times its depth and timber 15” is quoting a rule that has an index behind it, and the index is a strength one — how far a member can span before its own bending stress governs, at a depth it is allowed. It is the mass index with the free variable held fixed and the constraint solved the other way round.

That inversion is worth seeing because it explains why the two rules of thumb sound like they contradict each other. “Timber is a better beam material” and “timber does not span as far as steel” are both true, and they are answers to different questions: the first holds the stiffness and frees the depth, the second holds the depth and frees the span. The material has not changed and neither has the arithmetic; only which quantity was pinned.

Every rule of thumb in structural design is an index with its constraint left implicit, and the useful discipline is to ask, whenever one is quoted, which variable it assumed was free.

Where this model stops #

The shape has to actually be free. The whole derivation rests on being allowed to change the depth. A beam in a floor zone, a plate in a fixed thickness, a member matching an existing one — in every case the free variable is not free, the exponent collapses towards one, and timber’s advantage collapses with it. The index is a statement about a design freedom, not about a material.

One constraint at a time. Deflection and strength are different limits with different indices, and they rank differently. Real members are governed by stiffness and strength and stability, and the index that applies is the one belonging to whichever governs. A timber beam chosen on $E^{1/2}/\rho$ may then fail the strength check, and the two indices do not rank alike.

Buckling is not in it. A slender member has a third failure mode and its own index, and it is the one that rules out the naive answer in a great many cases. A very deep light beam wins on $E^{1/2}/\rho$ and then buckles sideways at a fraction of its capacity.

The material data is a single number for a thing that is a range. Timber’s modulus varies by a factor of three between species and by 20% within a grade, and it has three different moduli depending on direction. An index quoted to three figures on data like that is a statement about the arithmetic.

What the picture cannot show #

The chart plots eight points, one per material, as though each were a place. Every one of them is a cloud — a range of grades, treatments, orientations and qualities — and for some of them the cloud is larger than the distance to the next material. Steel’s is small. Timber’s is enormous. Concrete’s covers a factor of three in modulus.

Nor does the chart contain any of the reasons materials are actually chosen: whether they can be joined, whether they burn, whether they last, whether anyone in the region can work them, whether they can be made in the shape required. Those are not refinements of a material index; they are constraints that eliminate candidates before any index is computed, and in practice they do most of the eliminating.

The index’s real use is therefore narrower and more valuable than it looks. It does not choose a material. It says which property combination to compare on once the shortlist exists — and it says that the comparison a property table invites is, for almost every structural member, the wrong one.

The material that is not on the chart #

There is a structural material this table cannot rank at all, and its absence says something about the method’s limits.

Reinforced concrete is not a material; it is two materials arranged, and its performance depends on the arrangement rather than on either constituent. Concrete appears on the chart at $E = 30$ GPa and $\sigma = 30$ MPa, which describes it in compression and is a caricature of it in tension. The composite it forms carries tension at 500 MPa in some places and none at all in others, in a pattern the designer chose.

Every index above assumed a homogeneous member. As soon as a member’s material varies from point to point — reinforced concrete, a composite deck, a sandwich panel, a prestressed beam — the mass depends on how the materials were distributed, and distributing them is the design rather than an input to it.

That is not a defect in the index; it is the boundary of what it is a statement about. The chart ranks materials that a member could be made of. It has nothing to say about members made of arrangements, which is most of the ones on this site.

The generalisation #

The habit worth carrying is the derivation rather than any of the answers.

Write the objective. Write the constraint. Eliminate whatever the design is free to choose. What is left is a group of material properties, and it is the only group worth comparing on.

That procedure works far past materials. It is what produces the span-to-the-fourth relationship for deflection, what produces the cube in a pipe’s collapse pressure, what says a shape factor is a price quoted as a bonus. In each case a dimension was eliminated and an exponent appeared, and the exponent is the whole content of the result.

Which suggests the question to ask of any comparison: what was held constant, and what was allowed to vary? A comparison that has not answered both is not a comparison of anything.