Generator

The characteristic strength, which nothing was measured at

Rendered here at the parameters it defaults to, with every essay that calls it — which is the same list as the blast radius of changing it.

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.

9 essays call strength-scatter. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this generator changes every one of these figures.

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

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.

The dimension the capacity rides on is not a drawn dimension. Moment capacity of a 225 mm slab against the cover to the reinforcement. The line is very nearly straight, because capacity is A_s·f_yd·z and z is about 0.9d — so the capacity is proportional to a dimension that is not on the drawing. What is on the drawing is the overall depth, and the effective depth is what the cover, the link and half a bar diameter leave of it: 225 − 30 − 0 − 8 = 187 mm here. Every one of those three is a site tolerance rather than a design decision. Ten millimetres of bar position is 5.7% of this slab's capacity and 1.8% of a 600 mm beam's — the same workmanship costs 3.0 times as much in the shallow member, and the shallow member is the one whose steel is walked on before the pour.

Sections and stress

The dimension nobody can measure

Every flexural capacity in reinforced concrete is proportional to the effective depth, and the effective depth is not on the drawing. It is what is left of the thickness after a cover, a link and half a bar diameter have been taken off it — and each of those is a site tolerance. In a slab, ten millimetres of workmanship is six per cent of the strength.

The same steel, and a crack three times as wide. Calculated crack width against bar diameter, with the area of steel held at 1340 mm² per metre throughout — so the spacing changes with the square of the diameter and the amount of reinforcement does not change at all. The width runs from 0.173 mm at 8 mm bars to 0.372 at 25, a factor of 2.15 for identical steel. The reason is in the crack spacing: after a crack forms the bar has to re-anchor the concrete's tensile force before the next one can, and the length that takes is proportional to the bar's diameter. Of the 337 mm spacing drawn, 35% is the cover term and 65% is the bar term — and the cover term is the one that puts crack control and durability in opposition, because cover protects the bar and widens the crack that reaches it.

Sections and stress

The same steel, and a wider crack

A crack's width is the distance between cracks times the strain the steel carries over that distance. Neither of those is decided by how much reinforcement there is. The spacing is a bond length, so it goes as the bar diameter; the strain is set by the stress in the steel. Two arrangements of identical steel can differ by a factor of two in crack width, and the one that wins is the one with more, smaller bars.

$The ductility number depends on the ruler. Elongation after fracture against the gauge length it was measured over, in units of √S₀. A tensile specimen extends uniformly until the ultimate load and then localises into a neck, so the total is a strain (16 per cent here) plus a length (7.2 mm), and dividing a length by the gauge length is what makes the curve fall. At the two standard gauges the same steel reports 27.5 per cent over 5.65√S₀ and 21.8 over 11.3√S₀ — a ratio of 1.264, and 42 per cent of the first number is a property of the specimen rather than of the material.$ Materials

The ductility that depends on the ruler

Percentage elongation after fracture is the most quoted ductility measure in the subject and one of the least well defined. A specimen stretches uniformly until the ultimate load and then localises, so the number is a strain plus a length — and dividing a length by the gauge length makes the answer a property of the specimen.

$Independence, and the floor it never goes below. The reduction factor a column is allowed, two ways. The Eurocode storey rule falls to 0.70 and stops. The independence argument — n bays each with a mean and a standard deviation, summed — gives (1 + zv/√n)/(1 + zv), which falls faster and stops at 0.503: a floor with no n in it at all, decided only by how variable the load is and how far out the fractile is drawn. The mean is never reduced away, because every bay really does carry its mean. At ten storeys the two differ by 10.0 points.$ Equilibrium

The load that is never all there at once

A column at the bottom of twenty storeys is designed for the imposed load of twenty floors added up, and twenty floors do not reach their own worst day together. The reduction that follows is not a discount on the safety margin. It is the central limit theorem, and it has a floor it never goes below.

$Seventy per cent of the strength and ninety of the stiffness. Strength and modulus against age, each as a fraction of its own twenty-eight-day value. They do not move together: E follows f to the power 0.3, so at seven days the concrete has 78 per cent of its strength and 93 per cent of its stiffness, and at three days 60 and 86. A young structure is much nearer its final deflection than its final capacity. The 20 N/mm² a striking calculation asks for arrives at 2.2 days at 20 °C, 4.6 at five degrees and 1.7 at thirty-five.$ Materials

The strength it had on the day

Every concrete strength a design quotes is a twenty-eight-day cylinder value, and a structure is loaded long before that — formwork struck at three days, the next storey cast at seven, a prestressing force transferred at two. The number that existed at the moment the load arrived is a different one.

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.

Materials

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.

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.

Materials

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.

Materials

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.

The library, page 6 of 7 — where strength-scatter sits