The strength three cores revise

The strength that belongs to the test programme found that a characteristic strength estimated from a few specimens is lower than one estimated from many, not because the material is worse but because the estimate of its scatter is worse. With the scatter estimated from the same results, the factor applied to it is the Student $t$ quantile, which runs from 7.73 at two specimens to 1.73 at thirty; at four specimens the characteristic strength came out four N/mm² below what the same material, known exactly, would have given. A small test programme reports a worse strength.

That essay closed by naming the case where the effect is largest: the assessment of an existing structure, where the population is the structure itself and a handful of cores is the whole test programme. Drilling a core costs money and damages a member that is in use, so three is common and six is generous. Taken alone, three cores are penalised so heavily that an assessment can condemn a sound building on their scatter. But an existing structure is not alone. It was built to a specification, and the population of concretes to that specification is well known. This essay asks how much that knowledge is worth, and whether using it is safe.

What was ordered, and what was drilled #

The structure was built to a grade whose characteristic cylinder strength is 30 N/mm². Concrete produced to that grade has, across many sites and suppliers, a mean of about 38 N/mm² and a coefficient of variation of about 15 per cent; that is what a supplier aims for to pass its own conformity tests, and it is the population an engineer would assume before any core was taken. Three cores are then drilled from the members in question. They average 33 N/mm², with a coefficient of variation among them of 12 per cent: a little weaker than the grade’s population on average, and a little less scattered, which is about what an ordinary pour gives once the curing and the drilling have had their effect.

Strength is taken as lognormal, so the arithmetic is done on its logarithm. The specification supplies a prior on two things: the mean of log-strength, with the weight of a number of specimens $n'$, and its variance, with a number of degrees of freedom $\nu'$. Here $n' = 3$ and $\nu' = 10$ — a prior that knows the scatter of concrete fairly well and the mean of this concrete only loosely, because one pour can sit well away from its grade’s average. The cores add their own count to both:

$$n'' = n' + n, \qquad m'' = \frac{n' m' + n\bar y}{n''},$$

$$\nu'' s''^2 = \nu' s'^2 + (n - 1)s^2 + \frac{n' n}{n''}(\bar y - m')^2.$$

The last term is the one that makes this safe, and it will matter later: when the cores’ mean disagrees with the prior’s, the disagreement is added to the scatter. The characteristic value is the 5 per cent fractile of the predictive distribution, $\exp(m'' - t_{\nu''}\, s'' \sqrt{1 + 1/n''})$. With no prior at all — $n' = \nu' = 0$ — this is the lone estimate from the cores, and with a large prior and no cores it is the specification’s.

This is not a novel method. EN 1990’s Annex D gives exactly these formulae, with exactly these symbols, for the design of a structure assisted by testing, and the probabilistic model codes recommend them for assessment. What it is worth in practice is the question.

Three cores and a specification #

The hero figure draws the three answers as distributions. The specification alone is broad and centred above the cores — a mean of 38 with 15 per cent scatter, and a characteristic of 27.5 N/mm², as the grade intends. The cores alone are centred where the cores are, at 33, but their distribution has heavy tails: three results give a scatter estimate with two degrees of freedom, and the $t$ distribution with two degrees of freedom is wide. Its characteristic value is 21.9 N/mm², 27 per cent below the grade the building was ordered to.

Together they give 26.5 N/mm². The centre moves most of the way to the cores, because the cores say something specific about this concrete and the prior on the mean is weak. The scatter comes mostly from the prior, because three cores say very little about it and the prior says a good deal. The cores decide where the concrete is; the specification decides how much it is allowed to scatter. The result sits within half an N/mm² of 26.9, the characteristic of a population with the cores’ own mean and scatter, which is what an infinite number of cores would eventually say.

The prior is worth most when the cores are fewest #

The next figure follows each estimate as more cores are drilled.

The lone estimate climbs steeply from very little: 13.0 N/mm² from two cores, 21.9 from three, 26.0 from ten, 26.7 from thirty, approaching the population’s 26.9 as the $t$ factor approaches the normal one. The revised estimate is almost flat, at 26.3 to 26.7 for any number of cores, because the prior has already supplied what the first few cores could not. The two meet at about twelve cores and run together after.

The heavy prior is a warning. Given the weight of thirty specimens on its mean, the specification holds the estimate up near its own value: 28.7 from three cores, still 27.2 from thirty, above the population the cores actually come from at every count. A prior that heavy is a statement that this concrete is almost certainly the grade’s average concrete, which three cores already contradict.

The difference between the ordinary and the heavy prior is the difference between using the specification as what it is — a description of the population this concrete was drawn from — and using it as a description of this concrete. The first is honest about one pour being able to sit away from its grade’s average; the second is not.

A specification is worth a dozen cores or more #

The prior’s value can be stated as a number of cores.

Three cores with the specification reach 26.5 N/mm²; cores alone reach that only at twenty-one. Two cores with the specification reach 26.7, which alone takes thirty-seven. The worth falls as the cores multiply and settles at about fourteen, the count at which the lone estimate’s $t$ penalty has shrunk to what the prior’s own uncertainty costs. Beyond that the measure stops meaning much: both estimates have arrived near the truth, and the prior’s job is done.

For an assessment, the arithmetic is decisive. A core costs a drilling crew’s half-day and leaves a hole in a member that may be the one in question. Eighteen cores not drilled, for an estimate as good, is the difference between an assessment that can be afforded and one that cannot — and between a building condemned on three cores’ scatter and one shown to be adequate.

Scattered cores #

The cores’ scatter is where the lone estimate is most fragile.

Three cores that happen to agree closely, at 6 per cent, give 26.9 N/mm² alone and 26.9 with the prior: when the cores are consistent, the $t$ penalty acts on a small number and costs little. Three cores at 20 per cent give 16.6 alone and 25.7 with the prior; at 30 per cent, 11.7 and 24.2. The lone estimate reads three scattered results as a scattered concrete and multiplies that scatter by a large factor. The revised estimate reads them partly as an unlucky draw from a concrete whose scatter is known to be about 15 per cent, and lets them widen it only as far as their weight allows.

Three cores spread over 30 per cent are not rare. A core drilled through a bar, one taken near a construction joint, one from a member that was poorly compacted at its top — any of these puts a low result among two good ones, and a lone estimate then sinks by half. The strength it had on the day found that a concrete’s strength depends on more than its mix; a core’s strength depends on more than its concrete, and a prior that knows how much cores usually scatter is a defence against the one that went through a bar.

A weaker concrete, caught #

The obvious objection is that a prior built from the specification will flatter a concrete that did not meet it. The next figure tests that directly.

The cores now average 28.1 N/mm², a concrete whose own characteristic is 22.9 — below the grade of 30, a building that genuinely does not meet its specification. With the ordinary prior, three cores give 23.3, within 0.4 N/mm² of the truth. The prior has not pulled the estimate up towards 30, because the disagreement between the cores’ mean and the prior’s was added to the scatter, in the last term of the updating, and a wider scatter lowers the fractile. Cores that disagree with the specification make the revised estimate less confident, not more optimistic.

The heavy prior fails that test. Weighted as thirty specimens, it gives 27.4 N/mm² from three cores — 4.5 N/mm² above the truth, near enough the grade to pass an assessment it should fail. Even ten cores leave it at 25.1. The disagreement term is still there, but a heavy prior on the mean shrinks the estimate towards the specification by so much that the widened scatter cannot make up the difference.

The heavier the prior, the more it decides #

The last figure draws the prior’s weight as the variable, for three concretes.

As the prior’s weight grows, every concrete’s estimate is pulled towards the specification: the weak concrete’s up, from 18.6 alone through 23.3 at three to 25.7 at thirty; the strong one’s down, from 30.6 at three to 28.7 at thirty, far below its own population’s 36.3. At a weight of zero, every estimate is the lone, heavily penalised one, below its own truth. Somewhere between, at a weight of a few specimens, each estimate passes closest to its truth.

That few is the honest weight for a specification, and there is a reason it is small. A grade’s population includes every pour by every supplier; how far one structure’s concrete sits from that population’s mean is itself scattered by the coefficient of variation between pours, which is comparable to the scatter within one. The specification says little about this pour’s mean, and a lot about how much concrete scatters in general. Weighting the mean lightly and the scatter heavily, as $n' = 3$ and $\nu' = 10$ do here, is that statement in numbers.

What the estimate is then used for #

A characteristic strength in an assessment is not the end of the calculation; it is divided by a partial factor and fed into a check of the member’s resistance. The partial factor for concrete, 1.5 in new design, carries several allowances at once, and the strength that belongs to the test programme separated them: the scatter of the material, the gap between the strength of a specimen and the strength in a structure, and uncertainty in the geometry and in the resistance model. In an existing structure some of those can be measured. The geometry can be surveyed, and the dimension nobody can measure at the design stage — the effective depth, with its cover and its bars — can be found with a cover meter. The material’s scatter is exactly what the cores and the prior estimate.

Assessment codes therefore allow a reduced partial factor when the material has been tested, on the reasoning that part of what the factor covered is now known. That reduction is only honest if the characteristic value it is applied to has carried the testing’s own uncertainty, and the lone estimate from three cores carries it in a very blunt form: a $t$ factor of 2.92 that has nothing to do with the concrete and everything to do with the count. The revised estimate carries it in proportion — a factor of 1.91 on a scatter that the prior knows — and a reduced partial factor applied to it is applied to something that deserved it.

The same reasoning puts a limit on the cores’ use. They say how strong the concrete is now, which is usually stronger than at 28 days; the bigger one is the weaker one found that a large member fails at a lower nominal stress than a small specimen of the same concrete; and a timber beam’s strength under a long load is barely more than half its strength in a test, as the load that was left on too long found — a concrete has a smaller effect of the same kind. A characteristic strength from cores is the strength of cores. Getting from there to the strength a member can rely on is a second set of factors, which updating does not touch.

And a core is not free of the structure it comes out of. Drilling one through a slab beside a column removes concrete from exactly the place that the core drilled after the slab was built found most sensitive to it. The cores an assessment does not need to drill, because a prior carries part of their information, are holes that the structure does not need to carry.

By hand, for three cores #

The cores’ mean of 33 N/mm² with 12 per cent scatter has a log-mean of $\ln 33 - 0.12^2/2 = 3.489$ and a log-standard deviation of 0.120; the specification’s 38 at 15 per cent has 3.627 and 0.149. With $n' = 3$ and $n = 3$, the revised log-mean is the average, 3.558, so the median is $e^{3.558} = 35.1$ N/mm².

For the scatter, $\nu'' = 13$ and $13 s''^2 = 10 \times 0.149^2 + 2 \times 0.120^2 + 1.5 \times (3.489 - 3.627)^2 = 0.222 + 0.029 + 0.029 = 0.280$, so $s'' = 0.147$. The disagreement between cores and specification contributed the same amount to it as the cores’ own scatter did. The $t$ quantile for thirteen degrees of freedom at 95 per cent is 1.771, and $\sqrt{1 + 1/6} = 1.080$, so the characteristic is $35.1 \times e^{-1.771 \times 0.147 \times 1.080} = 35.1 \times 0.755 = 26.5$ N/mm².

Alone, the same cores have $\nu = 2$, $t = 2.920$ and $\sqrt{1 + 1/3} = 1.155$, so $33 \times e^{-0.0072} \times e^{-2.920 \times 0.120 \times 1.155} = 32.8 \times 0.667 = 21.9$ N/mm². The difference between the two is almost all in the $t$ quantile: 2.920 on two degrees of freedom against 1.771 on thirteen.

Where the updating stops #

The prior is assumed right about the scatter. If the concrete is scattered more widely than its grade’s population — a structure whose pours were done by different contractors over years — the prior’s ten degrees of freedom will hold the scatter estimate too low for longer than they should, and only many cores will widen it.

The cores are assumed to be a random sample. They rarely are: cores are drilled where they can be, away from bars and from the members’ critical regions, and an assessment that drills only where it is convenient samples the convenient concrete. The strength no specimen had found that the strength in a structure differs systematically from the strength of its specimens; a prior does nothing about a bias that every core shares.

The structure is one population. A building whose columns were cast from one supplier and whose slabs from another is two populations, and pooling their cores averages two concretes into one that exists nowhere. And EN 13791, which governs cores in Europe, adds conversion factors between core strength and the cylinder strength a design uses, which change the numbers here by a few per cent and change nothing about how the prior helps.

Still open: cores that are not independent #

Every core here is an independent draw from the structure’s concrete. Cores drilled a metre apart in one slab are not: they share a pour, a curing, the same morning’s water content, and they agree with each other more than two cores from different floors would. Three correlated cores carry less information than three independent ones — perhaps that of two — and an updating that counts them as three is overconfident about where the concrete is. Whether a correlated set of cores should be counted as fewer, by how much, and whether the prior’s protection against a weak concrete survives once the cores agree with each other for reasons that have nothing to do with the concrete, is the question that an assessment drilling its cores all from one bay leaves open.