The strength no specimen had

Every calculation in this collection has divided something by a strength. The strength was a number, and the material does not have a number.

It has a population. Cast thirty cubes from one mix, cure them together and crush them, and thirty different answers come back, spread over a range of perhaps a third of their mean. The number used in design is one particular point on the distribution those thirty results describe, chosen by a rule, and the choice is more consequential than almost anything else in a set of material properties.

Which free body produced the number #

There is no free body. This is the one essay in the collection whose quantity comes from a distribution rather than from an equilibrium, and it is worth naming that as a change of subject rather than sliding past it.

What replaces the free body is a rule for choosing a point on a distribution. The rule almost everywhere is the 5% fractile: the value 95% of the population exceeds. Fit a lognormal to a mean $m$ and a coefficient of variation $V$ — lognormal because a strength cannot be negative and a normal fitted to 20% scatter puts real probability below zero — and the fractile has a closed form:

$$x_k = m\,\exp\!\left(-\tfrac{1}{2}\sigma_{\ln}^2 - 1.645\,\sigma_{\ln}\right), \qquad \sigma_{\ln} = \sqrt{\ln(1+V^2)}$$

At $V = 0.15$ that is 77% of the mean. At $V = 0.30$ it is 60%. The whole of the gap between a material’s average behaviour and the number a designer is allowed to use is a function of one quantity, and that quantity is the spread.

Scatter beats mean, and it is not close #

The consequence is the most useful thing in this essay.

Halving the coefficient of variation from 0.20 to 0.10 takes the characteristic value from 71% of the mean to 84% — a gain of 19%, which is exactly what raising the mean strength by 19% and leaving the scatter alone would have bought.

Those two routes to the same answer have wildly different costs. Raising a mean concrete strength by 19% means more cement, a lower water-cement ratio, and a mix that is harder to place. Halving the scatter means better batching control, better curing, and more consistent testing — a process improvement rather than a material one, usually cheaper, and one that improves everything else at the same time.

It is also the route almost nobody proposes, because the specification names a strength and the scatter is nobody’s line item. The same asymmetry appears wherever this argument does: the mean is a property somebody sells and the spread is a property somebody controls.

Three specimens cannot see the tail #

The second consequence is about testing, and it catches people who reason from a small sample as though the sample were the population.

If the population’s scatter is known in advance, the estimator uses $z\sqrt{1 + 1/n}$ — 1.90 at three specimens against 1.645 at infinity, so a small sample costs little. If the scatter has to be estimated from the same $n$ results, the normal quantile is replaced by a $t$ quantile and the factor runs from 7.73 at two specimens to 1.73 at thirty.

At three specimens the characteristic strength of this material comes out at 18.1 N/mm² against 23.2 for a population known exactly — 22% lower, for a material that is identical. That is not a wider confidence interval; it is a lower number, entering a design as a strength.

That is why an approval by testing is expensive in a way that surprises people: the cost is not the tests, it is the number of them needed before the statistics stop taking the answer away.

It has a practical corollary that comes up on every project where something unusual is tested: a three-specimen programme on an excellent material can report a worse characteristic strength than a thirty-specimen programme on a mediocre one. The rule is not being unfair. It is refusing to believe a tail it has not seen.

The chain, which is the same argument about geometry #

The third consequence connects this essay to one about size.

A structure made of many similar elements fails at the weakest, and the distribution of the minimum of $n$ draws is $1 - (1-F)^n$. Its median is the $1 - 0.5^{1/n}$ fractile of one draw, which for 250 elements at a coefficient of variation of 0.18 is 61% of one element’s median.

Nothing about the material changes along that axis. The scatter is the whole of the mechanism: at zero scatter the line is flat, and a chain of any length is exactly as strong as one link.

That is the bigger one being the weaker one arrived at without mentioning size at all — and it is why the two explanations of the size effect are different theories rather than variants. The statistical one is this argument; the energetic one is about crack growth. They agree about the direction and disagree by up to 175% about everything else.

What the partial factor is not doing #

The last consequence is about the number that comes after the fractile.

Divide the characteristic value by a partial factor of 1.5 and the result sits at a fractile of $6.4\times10^{-6}$ — one specimen in 155,818. No test programme in the world has established anything about that part of a distribution, and no lognormal fitted to thirty results is a description of it.

So the factor is not covering the scatter, because the scatter was already spent getting to the 5% fractile. What it covers is a different list: the difference between a specimen and the material in the structure, the difference between a laboratory test and a real load, the geometric tolerances, and the possibility that the model used is wrong. Those are model uncertainties, and they are not distributed like a material property at all.

The clearest evidence for that reading is the variety of the factors. Concrete carries 1.5 and steel 1.0 in the same code, on materials whose scatters are 0.15 and 0.05 — a ratio of three where the scatters differ by three, which looks like agreement until the fractiles are computed. The characteristic values already differ by the scatter; applying a further factor in the same ratio applies the correction twice. What actually separates them is that concrete in a structure is not the concrete that was tested, and steel in a structure is.

Where the number comes from for other properties #

Not every material property is a strength, and the fractile rule is applied differently to each.

The modulus is used at its mean, because stiffness matters for load-sharing between parts and a conservative estimate for one member is unconservative for its neighbour.

The tensile strength of concrete — the property a flaw sets and the one with the widest scatter of any in common use — is quoted at three fractiles at once — a lower one for anchorage, a mean for deflection, an upper one for capacity design — because which end of the distribution is unsafe depends on the check.

The yield stress of steel is quoted at a low fractile for strength and at a high one for capacity design, for the same reason: a beam that is stronger than expected forces its connection to be stronger still.

That last case is worth pausing on because it compounds the problem. The stress at which nothing happens is a definition rather than a measurement for every material without a plateau — a line of slope $E$ from a strain of 0.002, cutting the curve wherever it does. So for those materials the population being fractiled is a population of results of a construction, not of an event.

Which materials have which spread #

The coefficient of variation is the quantity everything in this essay turns on, and its value is a property of the material and its manufacture rather than of the science.

Steel is the tightest: about 5% on yield stress for a rolled section, because it is made under control in a plant with continuous testing and a specification that the producer has to hit. A characteristic value 92% of the mean is the reward.

Concrete is around 10 to 20% depending on the batching, which puts its characteristic value between 71 and 84% of the mean — the whole range of the figure above, decided by how the material is made rather than by what it is.

Timber is 20 to 30% before grading and much less after it, which is the entire purpose of a grading system: it does not improve any piece of timber, it sorts the population into narrower ones. A grade is a scatter-reduction device, and a material with a direction needs it more than most because its properties depend on where the tree’s defects happen to be.

Masonry is the widest in ordinary construction, at 25% or more, because the unit, the mortar and the workmanship all vary and the third is not measured at all.

Read against the 19%-for-halving rule, that list explains a great deal about how the four materials are used. Steel is worked close to its mean and its members are slender; masonry is worked far below its mean, which is one reason a masonry wall’s calculated capacity looks so unlike what a wall obviously does.

It also explains why a stronger steel does not change the modulus but does change the scatter. Higher-strength grades are made to tighter specifications, so their characteristic values sit closer to their means, and part of the gain from specifying a stronger steel is not strength at all.

Where the scatter comes from #

It is worth separating the sources, because they respond to different interventions.

The material itself. Concrete’s aggregate, cement and water vary batch to batch; timber contains knots; steel’s chemistry varies within a specification band.

The specimen. Its size, its curing, its surface, its alignment in the machine.

The test. Its rate of loading, its temperature, the operator. Loading rate alone is worth several per cent on a yield stress and much more on concrete — a steel that is stronger in a millisecond is the same steel, measured differently.

Of these the second and third are usually the largest for well-made materials, which is why standardising a test does more for a characteristic value than improving a material does. It is also why a material tested one way and used another has a characteristic strength that describes neither.

Where the model stops #

The distribution is fitted, not observed. A lognormal through a mean and a coefficient of variation is a two-parameter description of thirty numbers, and the 5% fractile lies at the edge of the data. The fifth percentile of the fit and the fifth percentile of the sample are different things.

Strengths are not independent. The chain argument assumed the elements’ strengths were independent draws. Adjacent concrete in one pour, or adjacent bolts from one batch, are correlated, and a positive correlation makes the chain stronger than the formula says.

Nothing here is about consequences. A fractile is a probability of a value being exceeded; a design decision is about the cost of exceeding it. Two materials with the same fractile in a component whose failure is ductile and in one whose failure is sudden should not carry the same factor, which is why the property that appears in none of the equations is decisive here without ever being computed, and the treatment of that is a subject the fractile does not enter.

What the picture cannot show #

A distribution drawn on a page has smooth tails, and the tails are the part nobody has measured. Everything to the left of about the tenth percentile in the first figure is an extrapolation of a functional form, and the design value sits several times further out than that.

Nor does the picture show which population is being described. The thirty cubes were made from one mix on one day by one gang. The material in the structure is a year of mixes, several gangs, and weather — a wider population with a larger scatter and a lower fractile, described by no test that was performed.

The generalisation #

The habit worth carrying is that every material property in a calculation is a summary of a distribution, and which summary depends on which way the answer is unsafe.

A strength is taken low because a weak specimen is what fails. A modulus is taken at its mean because both directions are unsafe somewhere. A yield stress for capacity design is taken high because the thing being protected is the rest of the structure. Three different summaries of three distributions, each chosen by asking what would be bad.

That question generalises well past materials, and it is the same one that decides whether a load is factored up or down, whether a friction coefficient is taken at its lower or upper bound, and whether a stiffness should be over- or under-estimated in a system with two paths. In every case the arithmetic is the easy half. The judgement is knowing which tail of which distribution the structure is standing on.