Structural Engineering & Statics
where the load goes
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Characteristic strength — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The characteristic strength, which nothing was measured at. A lognormal population of strengths with a mean of 30 N/mm² and a coefficient of variation of 0.15. The characteristic value is the 5% fractile — 23.2 N/mm², which is 77% of the mean, and which need not be the strength of any specimen that was tested. Dividing it by 1.50 gives 15.5, and the shaded sliver below that is the fraction of the population that would fail to reach it: 6.4e-6, or one in 155,818. A factor applied to a fractile is not covering the scatter, because the scatter has already been spent getting to the fractile.

    The strength no specimen had

    A material property is written into a calculation as a number, and a material does not have one. It has a population of strengths with a mean and a spread, and the number used is a low fractile of that population — a value that need not have been measured, that most of the material exceeds, and whose distance below the mean is decided entirely by the scatter.

    part 1 · materials
  2. Three specimens cannot see the tail. The factor k applied to the sample's own scatter when a characteristic value is estimated from n specimens. With the scatter known in advance it is z·sqrt(1 + 1/n) and barely moves; with the scatter estimated from the same n results it is the Student t quantile instead, and it runs from 7.73 at two specimens to 1.73 at 30. At n = 4 the characteristic strength comes out at 23.8 N/mm² against 27.9 for a population known exactly — 15% lower, for a material that is identical. A small test programme does not report a worse estimate of the strength; it reports a worse strength.

    The strength that belongs to the test programme

    A characteristic strength is a fractile of a distribution, and a distribution estimated from four specimens is not the same object as one known exactly. The same material tested four times reports a strength 15 per cent below what it reports when its scatter is known — and nothing about the material differs.

    part 2 · materials
  3. Three cores and a specification say more together. The predictive distribution of a core's strength in an existing structure: from a concrete specified to a mean of 38 N/mm² and a coefficient of variation of 15 per cent, that prior carrying the weight of 3 specimens on its mean and 10 degrees of freedom on its scatter, before any core is taken (dotted); from three cores of mean 33.0 N/mm² and coefficient of variation 12 per cent alone (dashed); and from both (solid); each with its 5 per cent characteristic value marked. The specification alone gives 27.5 N/mm²; the cores alone, 21.9, because three results cannot pin down a scatter; together, 26.5.

    The strength three cores revise

    A characteristic strength estimated from three specimens and nothing else is heavily penalised, because three results cannot say how scattered a material is. An existing structure is never nothing else. It was built to a specified grade, from a population of concretes whose scatter is known in general, and three cores drilled from it revise that knowledge rather than replace it. Together they give a characteristic strength that three cores alone would need twenty to match. The same arithmetic catches a concrete much weaker than its specification, because cores that disagree with the prior widen it — unless the prior is given so much weight that it no longer listens.

    part 3 · materials
  4. Counted honestly, cores from one bay hand the estimate to the specification. The share of the structure's concrete that falls below the 5 per cent characteristic value estimated from three cores from one bay of a structure's concrete of mean 33.0 N/mm² and coefficient of variation 12 per cent, updating the specification — a mean of 38 N/mm² and a coefficient of variation of 15 per cent, carrying the weight of 3 specimens on its mean and 10 degrees of freedom on its scatter, averaged over 2,000 drillings, against the correlation between two cores from one bay: the cores counted as independent (dashed), counted by their correlation (solid), and three cores drilled from three bays (light); dotted, the 5 per cent an honest estimate gives. At ρ = 0.5, 6.2% and 6.5%, and 4.8%; at 0.9, 7.4% and 7.3%, and 4.8%.

    The cores that agree for the wrong reason

    Three cores drilled a metre apart in one slab share a pour, a curing and a morning's water, so they agree with each other more closely than they agree with the rest of the building. Counted as three independent specimens, they overstate what is known — and with nothing else to go on, the estimate built on them leaves a quarter of the building's concrete below a value meant to leave a twentieth. Counting them by their correlation repairs that. But once the specification is in the update, the honest count hands the estimate back to the specification, and a concrete weaker than it was meant to be hides behind it. The repair is not in the arithmetic. It is in where the cores are drilled.

    part 4 · materials

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