Structural Engineering & Statics
where the load goes
Concept

Partial factor — where it appears

A divisor applied to a material strength or a multiplier applied to a load, covering the difference between a tested specimen and a built structure. It is a statement about one quantity, so a check whose uncertainty is geometric rather than material is not covered by any of them.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside 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.

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.

materials · Characteristic strength 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.

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.

sections · Effective depth Ten decades of time for forty per cent of the strength. Strength as a fraction of the five-minute test value, against the length of time the load is held, over 12 decades of seconds. The relation is a straight line on this axis, which is the reason a single duration factor works at all: every decade of time costs about the same amount of strength. A load held for 60 years leaves 59% of the short-term strength, and the member need not have moved — this is not creep, and it is not fatigue, because nothing about the load varies. It is that a static stress slowly breaks fibres. The curve has an asymptote at 18% that nobody quotes: below about a fifth of its short-term strength a member has no time to failure at all, so there is a stress under which duration stops being a question rather than merely becoming a small one.

The load that was left on too long

A timber beam that carries a load for fifty years fails at about fifty-nine per cent of the stress the same beam carries for five minutes in a testing machine. Nothing about the load varies, the member need not have deflected, and the relation between the two is a straight line on a logarithmic time axis over ten decades.

materials · Duration of load 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 How long the pumps must run is set by how fast the water comes back. The floors that must be cast before the pumps stop under a 20 × 30 m basement dug 6 m below a water table at the surface, its 0.9 m slab and walls weighing 18.7 MN against 35.3 MN of uplift when the water is back, so that the weight stays ahead of the water at every instant, against the water's recovery time on a logarithmic scale, for a floor every 5, 7 and 10 days. If the water returns at once, five floors — the building has to be nearly heavy enough on its own. With a floor every 5 days: 3 at a 3-day recovery, 2 at 7, 0 at 14; a floor every 7 days: 4 at a 3-day recovery, 3 at 7, 1 at 14; a floor every 10 days: 4 at a 3-day recovery, 3 at 7, 2 at 14. Dashed, the closed form for weight added continuously at a floor a week, which runs about one floor below the steps: a floor arrives at the end of its cycle, not through it.

The water that comes back while the floors go up

A basement below the water table is built dry, inside an excavation kept pumped, and the building that will one day hold it down is not there yet. When the pumps stop, the water comes back over days and the frame goes up a floor a week, and the basement is safe only if the weight stays ahead of the water at every instant. How long the pumps must run is not set by how heavy the building is. It is set by one ratio — how much weight the frame adds while the ground refills — and for a basement that needs five floors on it to hold itself down, a seven-day recovery asks for three and a fourteen-day one for one.

equilibrium · Uplift The water that rises after the basement is finished pushes on its slab first. Section through a line of columns of a 20 × 30 m basement dug 9 m, its 0.9 m slab held down at six columns and by 8 storeys above, designed for a water table 3 m below the ground. The water table as designed is 3 m below the ground, a 6 m head under the slab; dashed, the same water risen 1.0 m. The slab bows up between its columns and walls under the uplift less its own weight (exaggerated). As designed, the building holds itself down at 0.65 of its weight's capacity, the slab is at 0.37 of its bending strength and its cracks are 0.140 mm against the 0.200 mm long set for every crack in concrete that must keep water out. With the water 1.0 m higher: 0.76, 0.45, and 0.176 mm against the same 0.200.

The basement that leaks before it floats

A water table that rises after a basement is finished is a load that grows through the building's life, and the building's flotation check is the last thing it troubles. A metre of rise adds a sixth to the uplift on the whole building but over a quarter to the net load on the slab, which carries the water less its own weight. The slab's cracks, which a watertight basement is designed to, are the first thing to run out of room — and the limit for the cracks that pass right through it tightens as the water rises.

equilibrium · Uplift 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 Under a light building the first thing the pump was holding up is the building. The hours after the pump stops at which each check on the basement is used up — the slab's 0.2 mm cracks and the building's flotation — against the number of storeys over it, with the water table risen 3.0 m and ground recovering in 12 hours; drawn at 48 hours where a check is never used up. The cracks' deadline is 9.5 h whatever the building. Flotation's is at once under 2 storeys, 1.0 h under 4 storeys, 4.7 h under 5 storeys, 10.1 h under 6 storeys, never under 8 storeys; with 5 storeys or fewer it comes first.

The building held down by its pump

A basement whose water table has risen above its design level is often given back its design head by a pump under the slab, and from then on it is safe while the pump runs. When the pump stops the water comes back on the ground's own clock, and the checks the pump was protecting are lost in an order the building's weight decides. Under eight storeys only the slab's cracks were ever at risk; under five, the building's flotation check goes first, four and three-quarter hours after the pump stops, and the slab's strength is never in question at all. A structure that was safe by its weight has become one that is safe by a machine, and the machine's reliability is now one of its loads.

equilibrium · Uplift

Named alongside it

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

Characteristic strengthReliabilityFractileHydrostatic pressureScatterSize effectTestingUpliftCrack widthEffective depthPore pressureQuality control

All concepts