Structural Engineering & Statics
where the load goes
Concept

Camber — where it appears

A curvature built into a member so that it takes the intended shape under a chosen load rather than when it is unloaded. It is fabricated against one stage of a sequence, so it is right on one day and wrong on every other, and choosing the day is a specification rather than a calculation.

Named by 11 essays across 2 fields — each of them below, with the objects they name alongside it.

The props decide where the stress ends up. Bottom-fibre stress in the steel of a 12 m composite beam carrying 6 kN/m of wet concrete and 9 kN/m afterwards. Unpropped, the bare steel takes the first stage alone and reaches 146 MPa; propped, the finished composite section takes everything and reaches 93 MPa — a ratio of 1.57. 62% of the unpropped beam's final stress was locked in before the slab was structural at all. The deflections differ by 1.73 times for the same reason, and no drawing of the finished beam distinguishes the two.

The section that changed while it was being loaded

A stress is computed from a moment and a section modulus. When part of the moment arrived while the section was a different shape, there is no single section modulus to divide by — the stresses add and the properties do not, and two identical finished beams can differ by half again in stress with nothing on the drawing to say which is which.

sections · Staged section The least reliable number in the material decides the answer, briefly. What a twenty per cent error in the concrete's tensile strength does to a computed deflection, against how far past cracking the beam is. Well past the cracking moment it does almost nothing — at 1.9 times M_cr the spread is 56 per cent — because the section is nearly fully cracked and the interpolation has run out. Just above cracking it does everything: at 1.19 times M_cr the same twenty per cent moves the deflection by 3658 per cent. Tensile strength is the property with the widest scatter and the least direct test, and a beam designed to sit near its cracking moment has put the answer on it.

The curvature nobody applied

Concrete shrinks as it dries, by about half a millimetre in every metre. In a symmetrically reinforced member that is a shortening and nothing else. In a member with more steel in one face than the other — which is every beam and every slab — the steel holds one side back and the section bends, with no load on it at all.

deflection · Shrinkage curvature 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 The check that depends on a date. Total deflection and the deflection occurring after the brittle finishes are built, for one 12 m beam, against the day those finishes go up. The total barely moves — the beam ends up where it ends up. The increment falls from 32 mm at a week to 14 mm at a year, because creep is fast at first and slow later and a partition built early inherits nearly all of it: 44% of the final creep has already happened by day 28. The span/500 limit is 24 mm and the span/250 limit is 48; this beam passes the first only after day 25. Camber subtracts from both terms of the difference and therefore changes the upper curve and not the lower one, which is the reason a cambered beam can satisfy every total-deflection check and still crack the wall.

The limit that depends on a date

Total deflection can nearly always be met, and on a long span it is met with camber. The limit that actually decides the member is the other one — the deflection occurring after the brittle finishes are built — and camber does nothing for it whatever, because it is subtracted from both terms of a difference. The same beam passes or fails on the day the partitions went up.

deflection · Incremental deflection The rotation a bearing sees is made before it arrives. The end rotation of a 12 m pretensioned beam, 300 mm wide and 700 mm deep, with a 160 mm in-situ slab 1.2 m wide, in 60 per cent humidity, from the prestress's release at three days to thirty years, with sagging positive. Release turns each end 4.8 milliradians upward; creep takes it to 7.9 by day 28, when the beam is set on its bearings. The slab brings it back by 0.9, the surfacing by 0.1, and thirty years of creep take it to 8.1 upward. The band after erection is the imposed load (0.9 down) and a night with the top cooler (0.6 down) above the line, and a sunny day with the top warmer (1.0 up) below it. Every value in the band is below zero: under every load it will carry, the beam's ends still point up, and a level bearing is turned the same way for its whole life.

The angle made in the casting yard

A bridge bearing is designed for the rotation of the beam it carries, and the rotation is listed as a sum of load, temperature, creep and a tolerance. Followed through the life of a pretensioned beam, the largest term is none of those. It is the upward turn the prestress gives the beam's ends in the casting yard, before any bearing exists, and under every load the beam will ever carry its ends still point up.

deflection · End rotation What a millimetre on each member does at mid-span. A Pratt truss of eight panels, 24.0 m long and 2.4 m deep, carrying 60 kN of dead load at each top node, drawn with its depth exaggerated. Each member is drawn as thick as, and labelled with, the distance mid-span moves for a millimetre's change in that member's length: its force under a unit load at mid-span. The middle top-chord members move it 2.50 mm per mm — the span over four depths — and the chord members fall off towards the supports: 1.25, 1.88, 2.50, 2.50, 1.88, 1.25 along the top chord. The diagonals move it 0.80 per mm and most verticals 0.50; three verticals — the one at mid-span and the two beside the supports — carry nothing under a load at mid-span and move it not at all (dashed). A truss's shape is its members' lengths, weighted like this.

The camber that lives in the member lengths

A beam is cambered by bending it; a truss cannot be, because it has no curvature to bend — its shape is nothing but the lengths of its members. So a truss's camber is a cutting list, and each millimetre on that list arrives at mid-span multiplied by the member's force under a unit load there: two and a half for the middle chords of a truss ten times as long as it is deep, nothing at all for some verticals. Build only the chords to their dead-load strain and a quarter of the sag stays, the webs' share. Cut every member to a millimetre of scatter and the mid-span misses by six, an eighth of the camber.

deflection · Camber Fit the counters last, and the tolerance locks in nothing. The largest force random length errors lock into an eight-panel Pratt truss with a counter-diagonal in each of its six interior panels, 24.0 m long and 3.0 m deep, carrying 60 kN of dead load at each interior bottom joint, against the errors' standard deviation: the median draw (solid) and the 95th percentile (dashed) with every member fitted in the shop, and the same with the counters left loose until the truss carries its dead load and then cut to the gap (on the axis). With everything fitted, 174 and 288 kN at 1 mm, 348 and 577 at 2 — straight lines, because the force is proportional to the misfit. With the counters fitted last, nothing at any tolerance: the Pratt truss without its counters is determinate, and a determinate truss takes any length error by moving.

The counter fitted after the load

A truss with a counter-diagonal in every panel cannot take a change of length for free, so a camber cut into its members might seem to lock force into it. One cutting list locks in nothing — every member shortened by its own dead-load stretch — and the simpler lists lock in little. What locks in force is the shop: a millimetre of error per member puts 174 kN into some member of a truss whose largest dead-load force is 471. Leave the counters loose until the dead load is on and cut them to the gap, and the same errors lock in nothing at all.

deflection · Camber A cambered girder is an arch on its slings. A 40 m precast girder of 11 kN/m, lateral stiffness 2.55 × 10⁵ kN·m², hung 0.90 m below its roll axis, picked 2 per cent of its length in from each end, with 100 mm of camber at mid-length, drawn in elevation with the camber exaggerated. The girder's centre of gravity is two-thirds of the camber above the line through its ends, 67 mm; its lifting points, on the arc 2 per cent in from each end, are 8 mm above it. It rolls about the line through the picks (dashed), so the camber has lifted its centre of gravity 59 mm towards the roll axis: the girder hangs as though its hook were that much lower.

The camber that lowers the hook

A camber is built into a girder so that it ends up level in the finished structure, and nothing about it matters until then — except on the day it is lifted. A cambered girder hanging from its ends is a shallow arch, its centre of gravity two-thirds of the camber above the line through its lifting points, and it rolls about that line as though its hook were that much lower. On a long precast girder already close to its lifting limit, a hundred millimetres of camber takes the factor of safety against roll from 2.3 to 1.4. The camber grows while the girder waits in the yard, so the same girder becomes harder to lift every week it is stored. Pick it a fifth of its length in from each end and the camber drops out of the problem altogether.

deflection · Camber Closing late beats jacking again on the same date. For the 60 m concrete rib rising 6 m under 60 kN/m, struck at 28 days, on ground that spreads 8 mm at once and as much again as it consolidates, 90 per cent in 10 years: the springing moment after a century, against a date in the arch's first years — left three-hinged and closed on that date (solid), or fixed at striking and jacked back to no moment on that date (dashed); dotted, fixed and jacked at striking only, −173. At three months, −124 against −140; at a year, −84 against −117; at ten years, −15 against −76. The difference is the memory of the years before: the jacked arch's early restraint goes on relaxing after the second jacking, and the open arch had none.

The arch that waits with its hinges open

A concrete arch fixed on the day it is struck has to restrain every millimetre its rib creeps and shrinks and its ground settles for the rest of its life, and jacking it again a year later cannot make it forget the year before. Struck as a three-hinged arch and left that way, it takes the fastest part of every movement for nothing, because a determinate arch has no moment to spend on a change of length. Closed at a year it ends the century with half the springing moment of the arch fixed at striking, and a third less than the arch jacked again on the same date. The price is a crown that has dropped 78 mm while it waited, and a rib that is no longer the shape of its own thrust.

deflection · Rib shortening The middle reaction is a dimension, read off in millimetres. The middle reaction under dead load of a Pratt truss continuous over two 24.0 m spans, 3.0 m deep, carrying 60 kN of dead load at each interior bottom joint, against the height of its middle bearing above the level drawn, cut to the continuous list (solid) and as two simple spans (dashed). At the drawn level the continuous list lands with 572 kN, the analysis's value, and the simple-span list with 480 kN, the two spans' own share. Every millimetre the bearing stands high adds 2.6 kN: 10 mm high, 598 kN; 10 mm low, 545; the simple-span list's 92 kN shortfall is the bearing 35 mm low.

The pier's share is cut into the members

A truss continuous over a middle support has one reaction that statics cannot find, and the lengths its members are cut to decide it. Cambered to land level under its dead load on three bearings, a two-span truss puts 572 kN on its pier; cambered span by span, as two simple trusses, the same members on the same bearings put 480 kN there and carry the dead load with no moment over the pier at all. The list chooses the moment diagram. And a millimetre of shop tolerance on every member moves the pier's share as much as a bearing set fourteen millimetres out of level, so a reaction the design depends on has to be set by a jack rather than by a drawing.

deflection · Camber 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.

CreepConstruction sequenceShrinkageServiceabilityComposite actionDifferential shorteningFabrication toleranceLack of fitThermal gradientToleranceCompatibilityDeflection

All concepts