Structural Engineering & Statics
where the load goes
Concept

Sampling — where it appears

Drawing a few specimens from a population in order to estimate its properties, such as cores from a structure or cubes from a pour. The fewer the specimens, the less is known about the population's scatter, and a characteristic value pays for that.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

materials · Characteristic strength 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.

materials · Characteristic strength 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.

materials · Characteristic strength An old joint passes the bolts that have lost their preload. The chance that an M20 grade 10.9 bolt holds the inspection torque of 647 N·m without its nut turning, against the tension actually in it as a fraction of the minimum preload, for a joint new and 1 year old, 5 years old and 20 years old. New, a bolt at half the minimum preload passes 0% of the time and one at the minimum 83%; 20 years old, the bolt at half passes 80% of the time and one at a quarter 9%.

The wrench that reads the rust

The obvious check on a preloaded joint that has been in service for years is a wrench on the nut, and it measures the wrong thing. The nut holds against the bolt's tension times a friction bracket, and twenty years of rust raise the bracket until a bolt with half its preload passes four times in five. The washer that could have said more remembers only the day it was crushed.

connections · Slip-critical The survey knows the creep once most of the floors are set. The doubt about the creep of a 40-storey building cast a floor every seven days, its creep known to ±30 per cent, as the half-width of its 95% band, after a survey of every floor set by then — each read to a standard error of 1.0 mm — against the floor it is made after: from the survey alone (drawn at 100% where it is larger), and combined with the design's ±30%; beside them, the share of the building's floors still to be cast, which are all a survey can correct. The survey alone beats the design's doubt only after floor 25, with 38% of the floors still to cast; the combined doubt halves after floor 31, with 23% still to cast.

The survey that arrives after the floors

A tall building's lower storeys start shortening months before its upper floors are cast, so a survey every ten floors should tell the creep the building really has and let the floors still to come be set for it. It does tell it, to a millimetre's precision, after the twenty-fifth floor. By then most of the floors whose height it was meant to correct are already set, and the floor that ends furthest from level is the one no survey can help.

deflection · Differential shortening

Named alongside it

The objects these essays reach for when they reach for this one.

Characteristic strengthFractileQuality controlReliabilityScatterTestingTolerancePartial factorBolt tensionCamberCoefficient of frictionConstruction sequence

All concepts