Structural Engineering & Statics
where the load goes
Materials

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.

Assumes The deflection that arrives three years late.

15 min read 5 figures Scale changes everythingThe load that will not hold still

The deflection that arrives three years late and the stress that leaks away both ran on a single creep coefficient, a number that multiplies the elastic strain to give the strain that follows it. Both essays treated that number as belonging to the concrete. A design code does not compute it that way. Its formula takes four inputs — the concrete’s strength, its age when loaded, the humidity of the air around it, and the size of the member — and two of those four are not about the concrete at all.

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.
Fig. 1 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 humidity, and the same member sealed so that it cannot dry. After a year the coefficients are 1.93 and 0.81; after fifty years, 2.45 and 1.28. The shaded difference, 48 per cent of the exposed member’s creep, is what drying adds.

Two mechanisms under one coefficient

Concrete under a sustained stress creeps by two routes, and they were separated experimentally long before any code formula was written.

Basic creep is what a specimen does when it is sealed, so that no moisture can enter or leave. It is a property of the cement paste: under stress, water held in the finest pores of the hydrated gel migrates and the gel’s layers slide, slowly, for years. It depends on the concrete’s strength and on how mature the paste was when the load arrived, and on nothing outside the specimen.

Drying creep is the extra creep a specimen shows when it is loaded and allowed to dry at the same time. It is not simply creep plus the shrinkage a drying specimen would show unloaded; a loaded specimen that dries creeps more than the sum of the two, an observation Pickett reported in 1942 and which has carried his name since. Moisture leaving the paste under stress lets it deform in a way that neither stress alone nor drying alone produces.

Eurocode 2’s creep coefficient carries both inside one factor. The notional coefficient is φ0=φRH β(fcm) β(t0)\varphi_0 = \varphi_{RH}\,\beta(f_{cm})\,\beta(t_0), and all of the member is in

φRH=1+1−RH/1000.1 h03\varphi_{RH} = 1 + \frac{1 - RH/100}{0.1\,\sqrt[3]{h_0}}

— with small corrections above 35 N/mm². The 1 is the part that does not depend on drying. The fraction is drying: it vanishes when the relative humidity is 100 per cent, and it is larger for a smaller notional size h0h_0.

The same member, sealed

The first figure holds everything fixed except whether the member can dry. Exposed to air at 50 per cent humidity, a member of notional size 150 mm reaches a creep coefficient of 1.93 after a year and 2.45 after fifty. Sealed, the same member reaches 0.81 and 1.28. Drying supplies 48 per cent of the exposed member’s long-term creep.

The sealed member’s creep also develops more slowly: after a year it has less than two-thirds of its final value, where the exposed member has four-fifths. In Eurocode 2’s formula the time a coefficient takes to develop grows with humidity, and it reaches its upper limit in saturated air, so the sealed member’s small creep also arrives late.

The practical sealed members are real: a pile below the water table, a foundation in wet ground, the inside of a mass pour. So is the reverse. A heated building’s interior is at about 50 per cent humidity for its whole life, and its slabs are among the driest concrete there is.

The notional size is the surface it dries through

h0=2Ac/uh_0 = 2A_c/u is twice the cross-sectional area divided by the part of the perimeter exposed to the air. For a slab open on both faces it is the slab’s thickness; for a square column it is half its width; for a member sealed on some faces it counts only the faces that can dry. It measures how far moisture has to travel to leave, and a larger h0h_0 is a member that holds its water longer.

The idea is the one that decides how quickly a steel member heats in a fire. There a surface divided by a volume sets the rate at which heat arrives; here the same ratio, inverted, sets the rate at which moisture leaves. Both are why a small member responds faster than a large one to the same environment, and in both the material is identical and the shape decides.

A thin member creeps more than a thick one of the same concrete. The creep coefficient after 50 years of a concrete of mean strength 38 N/mm² loaded at 28 days, against the member's notional size 2Ac/u, at relative humidities of 40, 60, 80, 100 per cent. At 40 per cent it runs from 3.31 at 50 mm to 2.01 at 1,000 mm; at 60 per cent it runs from 2.64 at 50 mm to 1.76 at 1,000 mm; at 80 per cent it runs from 1.97 at 50 mm to 1.52 at 1,000 mm; at 100 per cent it runs from 1.28 at 50 mm to 1.28 at 1,000 mm. The drier the air the more the size matters, and in saturated air it hardly matters at all: size enters only through drying.
Fig. 2 The fifty-year creep coefficient of the same concrete against notional size, at relative humidities of 40, 60, 80 and 100 per cent. At 40 per cent it falls from 3.31 at 50 mm to 2.01 at 1,000 mm; at 60 per cent from 2.64 to 1.76; at 80 per cent from 1.97 to 1.52. At 100 per cent it is 1.28 at every size.

Which free body produced the number

The free body is a slice of the member, and what crosses its boundary is water rather than force.

Moisture leaves concrete by diffusion, and diffusion has a clock that goes as the square of the distance travelled: the time for a drying front to reach a depth LL is of the order of L2/DL^2/D, with DD the moisture diffusivity. For ordinary concrete DD is around 10−10 m2/s10^{-10}\,\mathrm{m^2/s}. A slab 150 mm thick drying from both faces has 75 mm to empty, and 0.0752/10−100.075^2/10^{-10} is about 5.6×1075.6 \times 10^7 seconds — nearly two years. A member with a metre to empty would take a century by the same arithmetic, which is why the core of a large pier never dries in its service life at all.

That is the physical reason a code needs the member’s size, and it is also the measure of how approximate its formula is. Diffusion says the development time goes as the square of the size; Eurocode 2’s time constant grows linearly with h0h_0 and stops at a ceiling of about four years. The magnitude goes as a cube root, which is a fit rather than a derivation. Both were chosen to match test data over the sizes that were tested, and the direction of every effect in the figures is physics; the exponents are statistics.

The same slice also shows why sealing one face matters so much. Moisture that cannot leave through the underside of a slab on a steel deck has to travel the whole thickness to the top face instead of half of it, and by the diffusion clock that is four times as long. The doubling of h0h_0 that the formula gives a slab sealed on one face understates that, and it is on the safe side for the slab’s creep and the unsafe side for how long the slab keeps curling as its top dries.

The size enters as a cube root, so it matters most among thin members and flattens among thick ones: going from 50 mm to 150 mm does more than going from 300 mm to 1,000 mm. And the drier the air the more the size matters. At 40 per cent humidity the thinnest member creeps 1.65 times as much as the thickest; at 80 per cent, 1.30 times; in saturated air, not at all.

In wet air, size stops mattering

The same numbers read the other way round make the point the formula is built on.

With no drying, size stops mattering. The creep coefficient after 50 years of a concrete of mean strength 38 N/mm² loaded at 28 days, against the ambient relative humidity, for notional sizes of 50, 150, 300, 600, 1000 mm. The 50 mm member runs from 3.31 at 40 per cent to 1.28 at 100; the 150 mm member runs from 2.69 at 40 per cent to 1.28 at 100; the 300 mm member runs from 2.39 at 40 per cent to 1.28 at 100; the 600 mm member runs from 2.15 at 40 per cent to 1.28 at 100; the 1000 mm member runs from 2.01 at 40 per cent to 1.28 at 100. The curves close as the air gets wetter and meet at 100 per cent, where nothing dries and the notional size has left the coefficient.
Fig. 3 The fifty-year creep coefficient against relative humidity, for notional sizes of 50, 150, 300, 600 and 1,000 mm. At 40 per cent they run from 3.31 for the thinnest to 2.01 for the thickest. As the air gets wetter the curves close, and at 100 per cent every one of them is at 1.28.

The five curves meet at a single point, and they meet there because the size of a member is in the coefficient only through the drying term, and the drying term is zero when nothing can dry. A 50 mm panel and a 1 m pier of the same concrete, both in saturated air, creep alike. The size effect in concrete creep is not a size effect in the material. It is the time it takes water to leave.

That is also why a common explanation is wrong. Bigger members are sometimes said to creep less because they are better cured, or develop more strength in their cores. If that were the mechanism, the curves would not meet in saturated air; the thick member’s better curing would still be there. They do meet, in the model and in the tests the model was fitted to.

A thick member creeps later as well as less

Moisture leaves a thick member slowly, and the drying creep that depends on it arrives slowly with it.

A thick member creeps later, not only less. The creep of a concrete of mean strength 38 N/mm² loaded at 28 days at 50 per cent relative humidity, as a fraction of each member's own coefficient after a very long time, against time since loading, for notional sizes of 50, 150, 300, 600, 1000 mm. The 50 mm member has half its creep after 35 days and nine-tenths after 2.0 years; the 150 mm member has half its creep after 51 days and nine-tenths after 3.0 years; the 300 mm member has half its creep after 76 days and nine-tenths after 4.5 years; the 600 mm member has half its creep after 126 days and nine-tenths after 7.4 years; the 1000 mm member has half its creep after 159 days and nine-tenths after 9.4 years. Moisture leaves a thick member slowly, so its drying creep arrives years after a thin member's has finished.
Fig. 4 The creep of the same concrete at 50 per cent humidity as a fraction of each member’s final value, against time since loading. The 50 mm member has half its creep after 35 days and nine-tenths after 2.0 years; the 150 mm member after 51 days and 3.0 years; the 600 mm member after 126 days and 7.4 years; the 1,000 mm member after 159 days and 9.4 years.

The time constant in Eurocode 2’s formula grows in proportion to the notional size up to a ceiling. A slab 150 mm thick is nine-tenths of the way to its final creep in three years; a member with a notional size of a metre takes more than nine. The thick member has less creep and it has more of its creep still to come. At five years the thinnest member has 95 per cent of its eventual creep and the thickest 84.

That changes what a measurement means. A measured stiffness describes the structure that exists, and a deflection survey at five years of a building with thin slabs and deep transfer beams has caught the slabs nearly finished and the beams with a sixth of their creep still ahead. Extrapolating both from the same survey assumes they are the same distance along the same curve, which they are not.

One concrete, six coefficients

A single mix poured into the members of one building on one day is the case where all of this is simply true.

One concrete, one building, six creep coefficients. A concrete of mean strength 38 N/mm² loaded at 28 days at 50 per cent relative humidity, cast into six members of one building, with each member's creep coefficient after a year, marked by a tick, and after 50 years. The 150 mm slab open on both faces, notional size 150 mm: 1.93 and 2.45; the 200 mm slab on a steel deck, notional size 400 mm: 1.50 and 2.12; the 400 mm square column, notional size 200 mm: 1.80 and 2.35; the 800 mm square column, notional size 400 mm: 1.50 and 2.12; the 300 mm core wall, notional size 300 mm: 1.62 and 2.21; the 1.2 m transfer beam, notional size 800 mm: 1.22 and 1.93. The 150 mm slab open on both faces creeps 1.27 times as much as the 1.2 m transfer beam, from the same mix, poured on the same day and carrying the same stress, because its notional size is 5.3 times smaller and moisture leaves it faster.
Fig. 5 The same concrete at 50 per cent humidity in six members of one building, with each member’s creep coefficient after a year, marked by a tick, and after fifty years. A 150 mm slab open on both faces reaches 2.45; a 400 mm square column, 2.35; a 300 mm core wall, 2.21; a 200 mm slab on a steel deck and an 800 mm square column, both 2.12; a 1.2 m transfer beam, 1.93. The slab creeps 1.27 times as much as the transfer beam.

Two rows of that chart share a number, and the pairing is the lesson. A 200 mm slab on a steel deck and an 800 mm square column have the same coefficient, 2.12, because the deck seals the slab’s underside and doubles its notional size to 400 mm, which is the column’s. The same slab open on both faces would have a notional size of 200 mm and a coefficient of 2.35, which is the 400 mm column’s. What decides is the distance to a drying surface, not the member’s thickness.

A second pairing runs against a familiar picture of a tall building. The columns are the members that shorten most, because they carry the highest stresses, and the core is thought of as the stiff, massive element beside them. But the core’s walls are thin: a 300 mm wall open on both faces has a notional size of 300 mm and a coefficient of 2.21, while an 800 mm column has 400 mm and 2.12. Per unit of stress, the wall creeps more than the column, so the difference in shortening between them is smaller than the difference in stress suggests — and where a core wall is the more highly stressed of the two, larger than a single coefficient would give.

A factor of 1.27 between the slab and the beam sounds modest, and in the quantities that depend on it the difference compounds. The long-term deflection of a member under sustained load goes as 1+φ1 + \varphi, so the slab’s elastic deflection grows 3.45 times and the beam’s 2.93. The effective modulus a designer uses for the long term, E/(1+φ)E/(1+\varphi), is 29 per cent of the short-term value in the slab and 34 per cent in the beam. Where the two are connected — a slab spanning onto a deep beam, a column beside a core wall, a thin deck on thick webs — the difference is an imposed deformation between them that nobody applied, and an imposed deformation in concrete behaves differently from a load.

The same slab in a different room

Humidity moves the coefficient as much as size does, and the humidity is set by where the member is rather than what it is.

Take the 150 mm slab again. Inside a heated building, at 50 per cent, its fifty-year coefficient is 2.45. The same slab as the roof of a basement car park, open to outside air at about 80 per cent, has a coefficient of 1.76. Under the same sustained load its long-term deflection is 1+φ1 + \varphi times the elastic value: 3.45 times indoors and 2.76 times in the car park. The indoor slab deflects a quarter more over its life than an identical slab outdoors, and nothing about the slab, its load or its reinforcement differs.

That runs against an instinct that the harsh environment is the demanding one. For durability it is: the car park slab has the chlorides, the carbonation and the freeze–thaw. For creep the benign environment is the demanding one, because dry air is what drives drying creep, and the most comfortable room in a building is the one that pulls the most water out of its floor.

What a designer does with it

Three habits follow, and each is cheap.

Compute the notional size member by member, counting only the faces that dry. A slab on a steel deck, a wall cast against a sheet pile, a beam encased in a finish that seals it, and a column with a cladding tight against one face each have a larger h0h_0 than their dimensions suggest. The deck slab is the common case, and it moves the coefficient by about a tenth.

Choose the humidity for the exposure, not the climate. Fifty per cent is the usual value for a heated interior and eighty for outside air, and a building has both. A transfer structure in a basement and a roof slab above a heated floor are not in the same room even when they are in the same building.

And when a coefficient enters a difference, compute both ends of it. A slab spanning onto a deep beam, a column beside a core, a precast unit under an in-situ topping: each is a pair of members with different notional sizes, and the difference between their creep is the quantity that cracks a partition or tilts a floor. Using one coefficient for the whole building sets that difference to zero, which is the one value it is certain not to have.

Where the model stops

The two mechanisms share a clock. In this formula the drying and non-drying parts are multiplied by the same development function, which is the older formulation. The fib Model Code of 2010 writes basic and drying creep as two terms with two different time functions, because basic creep continues logarithmically for decades while drying creep levels off, and over very long periods the two formulations part company.

The humidity is an average. A member’s relative humidity is taken as one number for its whole life, where real exposure is seasonal outdoors and depends on heating and ventilation indoors, and the early months — when the drying is fastest — are often during construction, in whatever air the unfinished building held.

The notional size is one number for a whole section. A T-beam’s thin flange and thick web dry at different rates, and the section’s creep is not uniform across it. The formula averages them.

Creep is linear in stress. Above about 45 per cent of the strength it is not, and a load left on long enough moves into the regime where creep and strength interact.

And the coefficient is a prediction with a wide scatter. Code creep models are fitted to test databases and carry a coefficient of variation of twenty to thirty per cent, which is comparable to the whole spread between the members in the last chart. The shape of the dependence on size and humidity is well established; the value for a particular pour is not.

Still open: the slab that dries from one face

The notional size treats a member as drying evenly. A slab on a steel deck or a ground-bearing slab dries from its top face only, so its top shrinks and creeps more than its bottom, and it curls — upward at its edges, with a gradient no uniform coefficient can describe. That is the first question left open: creep and shrinkage through a section’s depth rather than as one number. After it come precast beams and the in-situ slab cast on them weeks later, one old and dry, the other young and wet, redistributing stress between them for years. Creep redistribution of moments in a frame built in stages, and the theorem that it moves the moments toward those of a frame built all at once. And prestress losses, where the tendon’s force is consumed by exactly the drying creep of the member it is in, and a thin-flanged box girder loses more than a solid slab of the same concrete.

The size effect is often taught as a correction to a material property. It is better taught the other way: the structure settles at a rate set by how fast it can give up its water, and the concrete’s own contribution is the part that would happen anyway, in the dark, sealed, for ever.

Where this goes next

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. 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. Two differences up the same building, peaking in different places. Differential shortening between a perimeter column and the core of a 40-storey building, plotted up the height. The part driven by load peaks at level 20 — exactly half way up, because a floor near the top has almost nothing built above it to shorten what is beneath, and a floor near the bottom has almost nothing beneath it to shorten. The part driven by shrinkage does not care what is above it at all and accumulates all the way to the roof. Their sum is worst at level 40, at 43 mm, which across a 9 m bay is a floor out of level by one in 208. The columns are shorter than the core Every column in a tall building gets shorter as the building is built on top of it, and the core beside it gets shorter by a different amount. The floors between them tilt by the difference — and the difference is largest exactly half way up, because a floor near the top has almost nothing built above it and a floor near the bottom has almost nothing beneath it. The fire is one curve, and the steel in it is several. Gas temperature and steel temperature against time in a standard fire. The gas curve is the same for every member in the compartment; the three bare steel curves are section factors of 76, 160 and 323 per metre, and they reach 558 °C at 17, 11, 8 minutes — a spread of a factor of two from geometry alone. The fourth curve is the middle section with 15 mm of board on it, which reaches the same temperature at 46 minutes. The kink near 735 °C on the bare curves is not a numerical artefact: steel's specific heat spikes there as its crystal structure changes, and the member spends several minutes absorbing heat at almost constant temperature. The temperature is a shape Two members of the same steel in the same compartment, under the same fire and the same load ratio, fail eight minutes apart. Nothing about the material differs and nothing about the fire does. What differs is a perimeter divided by an area, and it is the only number in the whole calculation that a designer chooses.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

AgeingCreepDeflectionEffective modulusImposed deformationModulusServiceabilityShrinkageSustained load