The strength that belongs to the test programme

A characteristic strength is a value no specimen had: the 5 per cent fractile of a population, computed from a mean and a scatter.

That calculation assumes the population is known. It never is, and what it costs to find out is the whole of this page.

What a sample costs #

The mean and the scatter both have to be estimated from the specimens that were tested, and the estimate of the scatter is the expensive one.

A small test programme does not report a worse estimate of the strength; it reports a worse strength. That is the finding, and the distinction matters because the two have opposite remedies.

An imprecise estimate is fixed by trusting the mean and accepting uncertainty. A lower characteristic value is what a designer must use, because the fractile has to be established with confidence and a small sample cannot establish it.

The gap closes slowly. Four specimens cost 15 per cent of the strength; ten cost about 6; thirty cost 2. The curve is steep at the left, so the difference between four and eight specimens is worth more than the difference between fifteen and thirty — and a test programme sized by convention rather than by that curve is usually sized in the steep part.

Which free body produced the number #

There is no free body here, and saying so is the point.

Every other page in this collection cuts something out, sums forces on it, and gets a number. This one is a statement about a population of specimens, and the object being reasoned about is a distribution rather than a structure.

That has a consequence for what the number means. A characteristic strength is not a property a piece of material has; it is a property of the process that made and tested a great many pieces. The bar in a particular beam is stronger or weaker than it, and which is unknowable.

So the whole reliability framework rests on a substitution: a value with a known probability of being exceeded replaces a value nobody can measure on the member in hand. The free body is the production line, and everything downstream — partial factors, load factors, target reliability — is bookkeeping about that substitution.

Why the penalty exists at all #

The Student factor looks like a piece of statistical bookkeeping, and the reason behind it is worth one paragraph because it explains why the penalty cannot be argued away.

The characteristic value is a point in the tail, and a tail’s position depends on the standard deviation. Estimating a standard deviation from four numbers is a poor business: the estimate itself has a distribution, and it is skewed — a small sample is more likely to under-estimate the scatter than to over-estimate it, because extreme values are rare and a small sample probably missed them.

So a small sample that reports a tight scatter is probably reporting a sample that happened not to contain an extreme, rather than a material that has none. The Student factor is the correction for that asymmetry, and it is large at small n because the asymmetry is.

That is why the penalty is not conservatism and cannot be removed by engineering judgement. Four specimens genuinely do not establish where the tail is, and a characteristic value is a statement about the tail — so the honest answer is either more specimens or a lower value, and the framework chooses the second when the first is unavailable.

Scatter is worth more than mean #

Two ways of improving a material compete, and one of them is cheaper than the other by an amount worth knowing.

Raising a mean strength means a better mix, a higher grade, a more expensive alloy. Cutting a scatter means better control: a consistent aggregate, a weighed batch, a covered stockpile, a calibrated machine.

The second is usually the cheaper of the two and is always the one that gets proposed second. That is a fact about how the industry is organised rather than about materials — a stronger grade is a purchase and a tighter control is a change of practice — and it is why the ready-mixed concrete industry’s improvement over fifty years has come far more from the second than from the first.

The asymmetry gets sharper as the scatter grows. At a coefficient of variation of 0.35 the characteristic value is barely half the mean, so half the material’s strength is being thrown away to pay for its own inconsistency — which is the situation for site-batched concrete, for reclaimed timber and for any material whose production is not controlled.

The chain, which is the same statistic #

The population’s tail also decides how a structure made of many pieces behaves.

The same population read as a chain rather than as a sample answers a different question, and the difference between the two questions is the whole of what a characteristic value is for.

That is the statistical size effect with no size in it. A large structure is a chain of many elements; a small one is a chain of few; and the large one is weaker in exactly the proportion the scatter dictates.

The mechanism is entirely the scatter. A material with no variability has a chain as strong as its links, and the whole of the size effect vanishes — which says that the size effect and the characteristic value are two readings of the same distribution rather than two phenomena.

The fall is logarithmic, which is the consolation. Going from one element to ten costs about 20 per cent; from ten to a hundred another 12; from a hundred to a thousand another 8. A structure is not proportionally weaker than a test specimen, and the reason design works at all despite the chain argument is that the logarithm is slow.

The scatter that is not in the material #

There is a second distribution in every member, and it is not covered by any material factor.

Neither the bias nor the scatter is a material property. The concrete and the steel are exactly as strong as they were tested to be; the member is 10.2 per cent weaker than the drawing says because a dimension nobody can measure after the pour came out different.

The material partial factor does not cover it, and no other factor is aimed at it either. What covers it, in practice, is that the material factors were calibrated against tests on real members rather than on coupons — so the geometric scatter is inside the calibration without appearing in the derivation.

That is worth knowing because it says where the framework can be trusted and where it cannot. A calculation that stays inside the calibrated range is protected against effects nobody wrote down; one that leaves it is not. A thin slab, an unusual cover, a member built to a tighter or looser tolerance than the tests: each moves the geometric distribution without moving anything the factors mention.

How the scatter gets into a design in the first place #

A designer never meets any of these distributions and meets their consequences constantly, and it is worth tracing the route.

A grade is a promise about a fractile. Specifying C30/37 concrete or S355 steel is specifying a characteristic value, which a producer guarantees by controlling a mean and a scatter that the designer never sees. The producer’s economics are entirely in that pair: a tighter scatter lets a lower mean meet the same grade, and a lower mean is cheaper.

Acceptance testing checks the promise on a sample. Cube tests, mill certificates, timber grading: each is a small sample from a large population, and each carries the sample-size penalty above — which is why acceptance criteria are written as rules about a run of results rather than about any one.

And an assessment inverts the whole thing. Taking cores from an existing structure means estimating a population from a handful of specimens, of a material whose production records are gone, in a structure that is the population. Every effect on this page is at its largest there: the sample is tiny, the specimens are correlated, and the geometric scatter is measurable rather than assumed.

So the framework is generous at design and harsh at assessment, which is the opposite of what the ages of the two structures would suggest and is a direct consequence of where the test programme sits.

What the partial factor is actually for #

The last figure’s arithmetic disposes of a common reading of the factor.

Dividing 27.9 by 1.50 gives 18.6, and the fraction of the population below 18.6 is 4.5 × 10⁻⁵. That is not “covering the scatter” — the scatter was already spent getting from 38 to 27.9.

The factor is doing something else: it is buying a reliability index. The design value has to be far enough into the tail that the probability of the resistance falling below the effect of the loads — which is itself a distribution — meets a target, typically about 10⁻⁶ per year for a structural element.

So $\gamma_M$ is calibrated rather than derived, and it is calibrated jointly with the load factors against a target probability. The two ends of the calculation are not independent, which is why mixing a characteristic strength from one code with a load factor from another is not conservative in any predictable direction.

One number, three different jobs #

It is worth separating the three quantities that get called “the strength”, because a great deal of confusion is a slippage between them.

The mean is what a producer controls and what a materials scientist quotes. It is the only one of the three that describes the material rather than a decision about it.

The characteristic value is a fractile, chosen by convention at 5 per cent, and is what a specification names. It depends on the mean, the scatter and — as this page has been about — the size of the sample that established them.

The design value is the characteristic divided by a partial factor calibrated against a target reliability jointly with the load factors. It is not a strength at all; it is a number arranged so that a comparison comes out right.

Only the first is a property of the material, and it is the one that appears in no calculation. That is a strange arrangement and it is the correct one: a design is a statement about a population of structures rather than about a piece of steel, and the population is what the other two describe.

What to carry away #

A characteristic value is a fractile of an estimated distribution, and the estimate’s quality is part of the answer rather than a caveat on it.

Four specimens cost 15 per cent of the strength. The penalty is steep at small sample sizes and shallow past about fifteen.

Cutting the scatter beats raising the mean, per unit of cost, and by a factor that grows as the scatter grows.

And the geometric scatter is real and uncovered. The member’s dimensions have a distribution too, with a bias, and no factor in the framework is aimed at it.

The material where all of it bites at once #

Every effect above is largest for one material, and it is worth naming because it explains why that material’s design rules look so different.

Timber has a coefficient of variation around 0.25 for bending strength — against 0.07 for structural steel and 0.15 for controlled concrete — so its characteristic value is a much smaller fraction of its mean. It is graded rather than manufactured, so the population is a sample of a forest rather than the output of a controlled process. Its elements are large relative to the test specimen, so the chain argument bites hard. And its strength depends on the duration of the load, which is a second distribution on top of the first.

The result is a design framework that looks unlike steel’s in every particular: a grade defined by a visual or machine assessment rather than by a chemistry, a size factor written into the strength, a duration factor, and a partial factor larger than any metal’s.

None of that is because timber is unreliable. It is because timber’s variability is visible — every one of these effects exists for steel and concrete and is small enough there to be absorbed into a calibration. Timber design is what the framework looks like when the statistics stop being negligible, which makes it the useful case to study rather than the awkward one.

Where the model stops #

The distribution is assumed lognormal. It is a reasonable fit for strength and it is a fit; the tail at the 5 per cent fractile is where the fit matters and where the data is thinnest.

The specimens are assumed independent. Cubes from one batch are not, so a test programme of thirty cubes from three batches has an effective sample size much smaller than thirty.

The chain assumes identical, independently distributed elements. Real elements share a batch, a supplier and a day’s weather, and correlated elements make a shorter chain than the count suggests.

Nothing here is about duration. A strength quoted from a five-minute test is a point on a curve, and for timber and concrete the duration correction is larger than anything on this page.

The 5 per cent fractile is a convention. Some materials and some codes use a different one — 1 per cent for a brittle material, a mean for a stiffness — and the number is not derivable from anything on this page.

And the tolerance distribution is assumed stationary. Workmanship varies between projects, between crews and over a pour, and the standard deviation used here is a national average standing in for a local one.

What it means for a specification #

Three practical consequences follow, and each is a decision made when a material is procured rather than when a member is designed.

Specify the grade, not the mix. A grade is a fractile and a producer meets it by controlling both parameters; a prescribed mix fixes the mean and leaves the scatter to whatever the process gives.

Size the test programme against the curve, not against convention. The penalty is steep below about ten specimens and shallow above fifteen, so a programme of six is in the expensive part and one of forty is buying very little.

And separate the two scatters. The material’s scatter is the producer’s to control and the geometry’s is the contractor’s, and no factor distinguishes them. A project that controls its concrete supply carefully and its cover badly has improved the half of the problem that was already better.

A characteristic value is a statement about a population, and three other essays here are about the ways one population is not another. The bigger one is the weaker one says the specimen size changes the answer; the strength it had on the day says the age does; and the load that was left on too long says the test duration does. All three are properties of the test rather than of the material, and none of them is carried by the number that reaches a design.

The ladder from here #

Later rungs on this anchor: the reliability calculation itself, with the resistance and the load-effect distributions convolved and a target index. Calibration of partial factors, and why a code’s factors are a set rather than a list. Bayesian updating from site tests, which is how a characteristic value is revised when a structure is assessed rather than designed. Correlated specimens and the effective sample size. Acceptance testing and the operating characteristic curve, which is the producer’s and consumer’s risk stated properly. And the assessment of existing structures, where the population is the structure itself and a handful of cores is the whole test programme — the case where every effect on this page is at its largest.

The 5 per cent fractile is a convention rather than a derivation, chosen in the 1950s and 60s as codes moved from permissible stress to limit states because it was a value that could be computed from a mean and a standard deviation without anybody having to agree on a target reliability. The target came later and the fractile stayed, which is why the two are separate numbers in every modern code and why the second is doing work the first is usually credited with.