Structural Engineering & Statics
where the load goes
Concept

Fractile — where it appears

A value with a stated probability of being exceeded, measured from the mean in standard deviations. It is what makes a characteristic strength and a characteristic load comparable, and the gap between a mean and a fractile is the whole of what a load reduction has to work with.

Named by 10 essays across 2 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 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.

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.

equilibrium · Live load reduction 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 The floor under the reduction rises with the correlation. The least factor a column carrying ever more floors may take, against the correlation between floors, for an office imposed load with a coefficient of variation of 0.6, its characteristic value the 95th percentile, loads normal: (1 + zv√ρ)/(1 + zv). With independent floors it is 0.503, the floor the independence argument found; at a correlation of 0.1 it is 0.660, at 0.3 0.775, at 0.5 0.855. EN 1991-1-1's storey rule falls to ψ₀ = 0.70 (dashed), and the two cross at a correlation of 0.157: floors that move together any more than that make the rule's tall-building reduction larger than the averaging can pay for.

The floors that fill up together

A column may be designed for less than the sum of its floors' imposed loads because the floors do not reach their worst day together — an argument that assumes each floor's load is independent of every other's. Let any two floors be correlated, and the reduction the averaging can pay for shrinks, and stops shrinking at a higher floor. At a correlation of 0.16 that floor reaches the 0.7 the storey rule falls to. One tenant on five floors halves what twenty floors can average away.

equilibrium · Live load reduction Split into what stays and what visits, the column needs far less. The factor a column may take on n floors' summed characteristic imposed loads, keeping the 50-year load at the 95th percentile, for office imposed load as a sustained part of mean 0.50 kN/m², renewed with each tenancy every five years on average, and intermittent loads of mean 0.2 kN/m² arriving 0.3 times a year on each floor for three days: floors let one by one, in tenancies of 5, and to one tenant; the one-number model with the same tenancies (dotted) and the storey rule (dashed). At 20 floors: 0.27, 0.37 and 0.60, against 0.75 for the one-number model and 0.73 for the rule. One floor's characteristic load is 3.24 kN/m².

The load that stays and the load that visits

An office floor's imposed load is two loads: a sustained one that arrives with a tenant and stays until the next, and intermittent ones — a party, a move, a crowd — that visit one floor for a few days. Simulated over fifty years, a floor's characteristic load is set mostly by its worst visit, and visits on different floors almost never coincide. So twenty floors let one by one need only 0.27 of their summed loads, in tenancies of five 0.37 and to one tenant 0.60, against the storey rule's 0.73. The rule is safe for offices by a wide margin. It stops being safe only where the load stays — an archive or a library let to one occupier — and there its column load is exceeded in a quarter of lives.

equilibrium · Live load reduction A quarter of the visits shared uses up the whole reduction. The factor needed by 20 floors carrying office imposed load — a sustained part of mean 0.50 kN/m² renewed every five years on average, and visits of mean 0.2 kN/m² arriving 0.3 times a year on each floor for three days, against the share of the visits made by the whole building at once: floors let singly (solid) and to one tenant (dashed), and the storey rule's 0.73 (dotted). With no share, 0.26 and 0.60; with a tenth, 0.51 and 0.67; with a quarter, 0.68 and 0.75; with all, 0.96 and 0.97. The rule's factor is reached at a share of 0.34 let singly and 0.21 to one tenant.

The visits that come to every floor at once

A floor's characteristic imposed load is set mostly by its worst visit — a party, a move, a crowd — and visits to different floors almost never coincide, which is why a column carrying twenty office floors needs only about a quarter of their summed loads. Let some of the visits be made by the whole building at once and nothing changes on any one floor; one floor cannot tell. The column can. A tenth of the visits shared takes the twenty-floor factor from 0.26 to 0.51, a quarter to 0.68, and a third uses up the whole of the storey rule's reduction. The rule fails in exactly the buildings whose use synchronises the crowd, and it fails most where the load mostly visits.

equilibrium · Live load reduction The corridor's share of a column decides its reduction. The factor on the summed characteristic loads that a column carrying n floors needs, against n, for office imposed load as the earlier essays drew it — a sustained part of mean 0.50 kN/m² and visits of mean 0.2 kN/m² a third of a year apart on each floor — and on the corridor a crowd of mean 1.00 kN/m² at one evacuation a year, on every floor at once, with 0%, 10%, 25%, 50%, 100% of the column's tributary area corridor; dashed, the storey rule. At 20 floors: 0.26, 0.30, 0.49, 0.90, 0.98, against the rule's 0.73. The lines are simulated, 4,000 lives at each point, and carry a few hundredths of sampling noise.

The column under the escape corridor

A column's imposed-load reduction is allowed by the number of floors it carries, on the reasoning that floors are not crowded at the same moment. A column whose tributary area includes escape corridor carries a part of every floor that is crowded at exactly the same moment, every time the building is evacuated. Its reduction follows how much corridor its area holds, not how many floors it carries — a tenth costs little, two-fifths uses up the whole of the storey rule's allowance, and a stair-core column needs almost none. Against the nominal loads a designer writes down, the margin that saves the columns at the corridor's edge has run out by the time the area is mostly corridor.

equilibrium · Live load reduction 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 One alarm in a few hundred that empties the building sets the column's load. The factor on the summed characteristic loads that a column carrying 20 floors needs, for office load and a corridor crowd at one alarm a year, evacuated three floors at a time, against the share of alarms that escalate to every floor at once (logarithmic), with 25% and 50% of the tributary area corridor; dotted, each with every alarm a full evacuation; dashed, the storey rule's 0.73. 25% corridor: 0.26 with none escalating, 0.27 at one in five hundred, 0.33 at one in a hundred, 0.38 at one in twenty, against 0.49 in full every time; 50% corridor: 0.29 with none escalating, 0.39 at one in five hundred, 0.59 at one in a hundred, 0.75 at one in twenty, against 0.90 in full every time.

The stair that empties three floors at a time

A column beside an escape stair carries the corridor crowd of every evacuation, and when every floor empties at once its floors are loaded together and earn almost no reduction. Tall buildings are usually evacuated in phases instead, three floors at a time, so that the stair is never asked to carry everyone. Under such a stair the column earns its reduction back and more — 0.29 of the summed floor loads, where the storey rule allows 0.73. But a phased strategy is a plan for the usual alarm, and the column's characteristic load belongs to the unusual one. If one alarm in a hundred escalates to a full evacuation, the factor climbs back to 0.59; one in twenty, and it passes the rule.

equilibrium · Live load reduction

Named alongside it

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

ReliabilityIndependenceCharacteristic valueImposed loadLive load reductionLoad combinationTributary areaCharacteristic strengthScatterTestingPartial factorQuality control

All concepts