Structural Engineering & Statics
where the load goes
Concept

Load combination — where it appears

A set of loads considered together, each at a factored value, representing one state the structure has to survive. Which combination governs is rarely the one with the largest total, because a structure responds to the arrangement of a load and imposed loads may be absent.

Named by 11 essays across 4 fields — each of them below, with the objects they name alongside it.

The envelope is not a state of the structure. Every arrangement of the imposed load on three spans — 8 of them, since each span is loaded or not — drawn faintly, with the greatest sagging and greatest hogging at each station drawn over them. Each faint curve is a real state of equilibrium and satisfies the free-moment identity exactly: mid-span ordinate minus the mean of the end moments is wL²/8, to 3e-16 of it. The envelope satisfies it nowhere, missing by up to 23% — because it is assembled from different load cases at different stations and no arrangement of load produces it. seven of the 8 arrangements are needed to build it; the rest never govern anywhere.

The envelope is not a structure

A continuous beam whose imposed load may sit on any span has eight load cases, and every one of them is a genuine state of equilibrium. The curve the design is made against is not one of them — it is assembled from different cases at different stations, and it fails the identity all eight satisfy exactly.

equilibrium · Load arrangement The load that makes itself worse. The amplification of a deflection against the ratio of applied load to buckling load. A structure at half its buckling load deflects twice as far as first-order analysis predicts, and the curve runs away well before the load is reached.

The load that is really a lean

No frame is ever plumb. The columns are out of upright by something like a three-hundredth, and every tonne of gravity load standing on that lean has a horizontal component. The force that represents it is not a safety allowance — it is an exact statics substitution for a geometry nobody drew.

equilibrium · Notional load The gap is a sum of five things and only one of them is computed. What a 30 mm movement joint is asked to accommodate, by three combination rules. The top bar is every term at its extreme, added: 33.2 mm, which assumes the hottest day, the fullest floor, the whole of the shrinkage and the worst-placed wall arrive together. The chance of that is about 1.5%. The bottom bar treats them as independent and asks for 16.0 mm. The middle bar is the rule used for actions and almost never for movements — one term at its full value and the rest at their coincidence factors — and gives 25.5 mm. The segments across the top bar are the terms themselves, and the ordering is the finding: the largest is tolerance at 10.0 mm, which is not a structural quantity at all, and the smallest is deflection at 3.2 mm — the only one anybody computes carefully, and 10% of the total.

The gap nobody computed

A movement joint is sized by adding up everything the structure will do to it, and the deflection calculation — the only term anybody computes carefully — is usually the smallest one in the list. The largest is a construction tolerance, which is not a structural quantity at all, and the sum of the extremes is nearly twice what treating them as independent would ask for.

deflection · Movement budget 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 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 The interior panel is designed by the case that designs nothing else. The shear across the column-web panel at each kind of joint of a frame of two 8 m bays and two 4 m storeys with fixed bases, 457 mm beams and 254 mm columns with 8.6 mm webs, under five load cases, the worse direction of sway taken where there is one. Full gravity: roof corner 106, roof interior 0, floor exterior 143, floor interior 0 kN; One bay loaded: roof corner 115, roof interior 117, floor exterior 150, floor interior 154 kN; Sway: roof corner 25, roof interior 40, floor exterior 53, floor interior 89 kN; Gravity + sway: roof corner 131, roof interior 40, floor exterior 196, floor interior 89 kN; Pattern + sway: roof corner 141, roof interior 157, floor exterior 203, floor interior 242 kN. The floor interior panel is empty under full gravity and carries 89 kN under gravity with sway, the case that governs the members; under one bay loaded with sway it carries 242 kN — 2.7 times as much, more than any exterior panel's 203 kN, and 70 per cent of the web's shear resistance.

The panel that gravity leaves empty

A corner joint's panel carries the difference between a beam's flange couple and half a column's shear, and full gravity load is what makes that large. An interior joint has a beam on each side, and under full gravity their two couples cancel: the panel between them carries nothing at all, in the load case that sizes every member around it. It fills under the two loads that are not symmetric. A pattern load hands it one beam's moment, and sway hands it two beams' moments turning the same way. Together they put more shear through it than through any exterior joint, in a combination that governs no member.

internal-forces · Corner moment 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 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.

Imposed loadCharacteristic valueFractileIndependenceLive load reductionTributary areaReliabilityServiceabilityCreepFree bodyLoad pathPattern loading

All concepts