Structural Engineering & Statics
where the load goes
Series

Creep — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The deflection that arrives years late. The multiplier on a concrete member's deflection under a sustained load, against time. The elastic deflection arrives on the day the load does and is the 1.0 at the left. After a year it has been multiplied by 3.00, after five years by 3.29, and it approaches 3.38. Nothing has been added to the load and nothing about the strength has changed: this is a serviceability failure arriving on a structure that passed every strength check on the day it was built.

    The deflection that arrives three years late

    A concrete beam that passes every check on the day it is built goes on deflecting for a decade, and ends up three times where it started. Nothing about the load changed, and nothing about the strength was ever in question.

    part 1 · materials
  2. The stress that leaks away. A restrained shrinkage strain of 320 microstrain in concrete of modulus 34000 N/mm². Ignoring creep it produces 10.88 N/mm², which is above the tensile strength of 3.8 and predicts that every restrained concrete member ever cast has cracked. Counting creep by the superposition integral leaves 1.72 N/mm² after 55 years, and the one-line age-adjusted shortcut at the usually quoted ageing coefficient of 0.8 leaves 3.05. The two disagree — this creep function implies an ageing coefficient of 1.67, not 0.8 — and both are below the tensile strength, so the conclusion turns on counting creep at all rather than on how it is counted.

    The stress that leaks away

    Creep makes a load's deflection grow and an imposed strain's stress shrink, and the second is why restrained concrete does not crack as often as an elastic calculation says. The same material property runs both ways, and which way it runs depends on whether the structure was given a force or a movement.

    part 2 · materials
  3. Half of the creep is the member drying out. The creep coefficient of a concrete of mean strength 38 N/mm² loaded at 28 days, in a member of notional size 150 mm, against time since loading: exposed to air at 50 per cent relative humidity, and the same member sealed so that it cannot dry. After a year the exposed member's coefficient is 1.93 and the sealed member's 0.81; after 50 years they are 2.45 and 1.28. The shaded difference, 48 per cent of the exposed member's final creep, is what drying adds. The sealed member's coefficient has no size in it; the drying part is where the member comes in.

    The creep that belongs to the member

    A creep coefficient is quoted for a concrete, and half of it is not a property of the concrete. It is the member drying out, and drying goes through the surface — so one mix creeps a quarter more in a thin slab than in a deep beam, gets there years sooner, and in saturated air forgets its size altogether.

    part 3 · materials
  4. The humidity through a slab's depth, and the strain it asks for. A slab 200 mm thick drying through its top face only into air at 50 per cent, with its underside sealed by a steel deck. Left, the pore humidity against depth at 28 days — 50 per cent at the surface and 100 at the base; 90 days — 50 per cent at the surface and 100 at the base; 1 year — 50 per cent at the surface and 99 at the base; 5 years — 50 per cent at the surface and 74 at the base; 20 years — 50 per cent at the surface and 51 at the base. Right, the free strain each of those profiles asks for, taken as shrinkage at the local humidity. At 28 days the top wants to be 461 microstrain shorter than it was and the bottom 0; at 20 years the two are 461 and 456, and the gradient that produced the curl has gone. Not one of the profiles is straight, and a section that stays plane cannot deliver any of them.

    The slab that dries from one face

    A creep coefficient is one number for a member, and a member drying through one face does not have one. Give every depth its own humidity and the section a strain profile it cannot deliver, and two things follow that no single coefficient contains — a six-metre slab lifts eleven millimetres at its edges with nothing on it, and its top surface is past cracking before it has been loaded.

    part 4 · materials
  5. Three moment diagrams for one pair of beams. Two 15 m spans carrying 12.0 kN/m, drawn sagging downward. As built they are simple spans: 337.5 kN·m at each midspan and nothing over the middle support. Built monolithic they would carry 337.5 kN·m of hogging over the support and 168.8 at midspan. Loaded at 28 days and made continuous at 60, creep takes them 46 per cent of the way from the first to the second: 155.2 kN·m over the support and 259.9 at midspan, after twenty years in which nothing about the loading changed. The support had no moment on the day it was cast and no drawing of the finished structure shows why it has one now.

    The support that had no moment when it was cast

    Two beams are set on their bearings, carry their own weight for a month, and are then stitched together over the middle support. Nothing about the loading changes afterwards. Twenty years later the stitch is carrying 155 kilonewton-metres, because the concrete went on creeping and the joint would not let it — and how much arrives is decided by a crane schedule.

    part 5 · materials
  6. Where a tendon's force goes, over fifty years. The loss of stress in a tendon stressed to 1300 N/mm² and released at 7 days into a member of notional size 300 mm at 70 per cent humidity, with the three causes stacked. At 28 days the total is 61.9 N/mm²; at a year 131.5; at fifty years 190.8, which is 14.7 per cent of what the tendon started with. Creep supplies 107.7 of that, drying and autogenous shrinkage 50.6, and the steel's own relaxation 32.5. Half the loss has happened by 119 d and nine-tenths by 8 y.

    The prestress the member takes back

    A tendon is stretched, locked off against the concrete, and then has to hold that extension while the concrete underneath it shortens by itself. Fifteen per cent of the force goes, most of it to creep, and how much goes is decided by the shape of the member and the air it stands in rather than by anything about the steel.

    part 6 · materials
  7. Thirty years later, two concretes against one. Stress down the composite section after thirty years — a 160 mm slab cast 6 weeks after a 700 mm pretensioned beam — computed twice: with the slab as a second, younger concrete that creeps and shrinks by its own laws, and with it given the beam's concrete and age. With two concretes the slab ends at −0.15 N/mm² at its top and −0.30 at its bottom, the beam at −4.03 at its top and −7.43 at its soffit. With one, the slab is at −0.29 and −1.06, the beam at −2.22 and −8.37. The slab's own shrinkage has taken its compression away and handed it to the top of the beam, and the soffit — the fibre the prestress was designed to keep in compression — has lost 0.94 N/mm² of it. Compression is negative.

    The slab that shrinks onto a finished beam

    A precast beam with an in-situ slab cast on it is one member made of two concretes, and they do not age together. The slab's shrinkage is nearly the same whenever it is poured; what changes is how much shrinking the beam has left to share it with. Cast the slab at six weeks and it takes a ninth of the soffit's precompression away over thirty years. Cast it at a year and it takes a quarter, and goes into tension itself.

    part 7 · materials
  8. The date the joint is cast decides the sign. The moment at the pier after thirty years, for two 12 m pretensioned beams, 300 mm wide and 700 mm deep with a 160 mm slab, made continuous over the pier by a joint cast with the slab, against the beams' age when the joint and slab are cast; sagging positive, with its three parts dashed. Cast at 7 days the joint ends at +320 kN·m; at 28, +214; at 90, +59; at a year, −164. It exceeds the joint's cracking moment of 159 kN·m for any joint cast before about 45 days, and it changes sign at about 130 days. From 7 days to a year the prestress's share falls from +555 to +204, because an old beam has made most of its upward creep before it is joined; the differential shrinkage's grows from −57 to −285, because an old beam has finished its own shrinking and the slab's is then all difference; the dead load's eases from −178 to −83. The first two move the joint the same way as the beams age.

    The pier that bends the wrong way

    Two precast beams are made continuous over a pier by a joint cast with the deck, and the joint is designed for the hogging moment a continuous beam has there. Thirty years later it is sagging, by more than the moment that cracks its underside, because the beams were still cambering upward when they were joined. Whether that happens is decided by two dates — when the beams were cast and when the joint was — and the deck's shrinkage, which pulls the other way, is not enough to stop it.

    part 8 · materials

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