Creep and shrinkage of concrete


Creep and shrinkage of concrete are two physical properties of concrete. The creep of concrete, which originates from the calcium silicate hydrates in the hardened Portland cement paste, is fundamentally different from the creep of metals and polymers. Unlike the creep of metals, it occurs at all stress levels and, within the service stress range, is linearly dependent on the stress if the pore water content is constant. Unlike the creep of polymers and metals, it exhibits multi-months aging, caused by chemical hardening due to hydration which stiffens the microstructure, and multi-year aging, caused by long-term relaxation of self-equilibrated micro-stresses in the nano-porous microstructure of the C-S-H. If concrete is fully dried, it does not creep, but it is next to impossible to dry concrete fully without severe cracking.
Changes of pore water content due to drying or wetting processes cause significant volume changes of concrete in load-free specimens. They are called the shrinkage or swelling. To separate shrinkage from creep, the compliance function, defined as the stress-produced strain caused at time t by a unit sustained uniaxial stress applied at age, is measured as the strain difference between the loaded and load-free specimens.
The multi-year creep evolves logarithmically in time, and over the typical
structural lifetimes it may attain values 3 to 6 times larger than the initial elastic strain. When a deformation is suddenly imposed and held constant, creep causes relaxation of critically produced elastic stress. After unloading, creep recovery takes place, but it is partial, because of aging.
In practice, creep during drying is inseparable from shrinkage. The rate of creep increases with the rate
of change of pore humidity. For small specimen thickness, the creep during drying greatly exceeds the sum of the drying shrinkage at no load and the creep of a loaded sealed specimen. The difference, called the drying creep or Pickett effect, represents a hygro-mechanical coupling between strain and pore humidity changes.
Drying shrinkage at high humidities is caused mainly by compressive stresses in
the solid microstructure which balance the increase in capillary tension and surface tension on the pore walls. At low pore humidities, shrinkage is caused by a decrease of the disjoining pressure across nano-pores less than about 3 nm thick, filled by adsorbed water.
The chemical processes of Portland cement hydration lead to another type of shrinkage, called the
autogeneous shrinkage, which is observed in sealed specimens, i.e., at no moisture loss. It is caused partly by chemical volume changes, but mainly by self-desiccation due to loss of water consumed by the hydration reaction. It amounts to only about 5% of the drying shrinkage in normal concretes, which self-desiccate to about 97% pore humidity. But it can equal the drying shrinkage in modern high-strength concretes with very low water-cement ratios, which may self-desiccate to as low as 75% humidity.
The creep originates in the calcium silicate hydrates of hardened Portland cement paste. It is caused by slips due to bond ruptures, with bond restorations at adjacent sites. The C-S-H is strongly hydrophilic, and has a colloidal microstructure disordered from a few nanometers up. The paste has a porosity of about 0.4 to 0.55 and an enormous internal surface area, roughly 500 m2/cm3. Its main component is the tri-calcium silicate hydrate gel. The gel forms particles of colloidal dimensions, weakly bound by van der Waals forces.
The physical mechanism and modeling are still being debated. The constitutive material model in the equations that follow
is not the only one available but has at present the strongest theoretical foundation and fits best the full range of available
test data.

Stress–strain relation at constant environment

In service, the stresses in structures are < 50% of concrete strength, in which case the stress–strain relation
is linear, except for corrections due to microcracking when the pore humidity changes. The creep may thus be characterized by the compliance function . As increases, the creep value for fixed diminishes. This
phenomenon, called aging, causes that depends not only on the time lag but on both and separately. At variable stress, each stress increment applied at time produces strain history. The linearity implies the principle of superposition. This leads to the stress–strain relation of linear aging viscoelasticity:
Here denotes shrinkage strain augmented by thermal expansion, if any. The integral is the Stieltjes
integral, which admits histories with jumps; for time intervals with no jumps, one may set to obtain the standard integral. When history is prescribed, then Eq. represents a Volterra integral equation for. This equation is not analytically integrable for realistic forms of,
although numerical integration is easy. The solution for strain imposed at any age
is called the relaxation function.
To generalize Eq. to a triaxial stress–strain relation, one may assume the material to be isotropic, with
an approximately constant creep Poisson ratio,. This yields volumetric and deviatoric stress–strain
relations similar to Eq. in which is replaced by the bulk and shear compliance functions:
At high stress, the creep law appears to be nonlinear but Eq. remains applicable if the
inelastic strain due to cracking with its time-dependent growth is included in. A viscoplastic strain
needs to be added to only in the case that all the principal stresses are compressive and the smallest in
magnitude is much larger in magnitude than the uniaxial compressive strength.
In measurements, Young's elastic modulus depends not only on concrete age but also on the test duration because the curve of
compliance versus load duration has a significant slope for all durations beginning with
0.001 s or less. Consequently, the conventional Young's elastic modulus should be obtained as,
where is the test duration. The values day and days give good agreement with the
standardized test of, including the growth of as a function of, and with the widely used empirical
estimate . The zero-time extrapolation happens to be approximately age-independent, which makes a convenient parameter for defining.
For creep at constant total water content, called the basic creep, a realistic rate form of the uniaxial compliance function was derived from the solidification theory:
where ; =
flow viscosity, which dominates multi-decade creep; = load duration; = 1 day,, ; = volume of gel per unit volume of concrete, growing due to hydration; and = empirical constants. Function gives age-independent delayed elasticity of the
cement gel and, by integration,.
Integration of gives as a non-integrable binomial integral, and so, if the values of are sought, they must be obtained by numerical integration or by an approximation formula. However, for computer structural analysis in time steps, is not needed; only the rate is needed as the input.
Eqs. and Asymptotically for both short and long times,, should be a power function of time; and 2) so should the aging rate, given by ) .

Creep at variable environment

At variable mass of evaporable water per unit volume of concrete, a physically realistic constitutive relation may be based on the idea of microprestress, considered to be a dimensionless measure of the stress peaks at the creep sites in the microstructure. The microprestress is produced as a reaction to chemical volume changes and to changes in the disjoining pressures acting across the hindered adsorbed water layers in nanopores, confined between the C-S-H sheets. The disjoining pressures develop first due to unequal volume changes of hydration products.
Later, they relax due to creep in the C-S-H so as to maintain thermodynamic equilibrium with water vapor in the capillary pores, and build up due to any changes of temperature or humidity in these pores. The rate of bond breakages may be assumed to be a quadratic function of the level of microprestress, which requires Eq. to be generalized as
A crucial property is that the microprestress is not appreciably affected
by the applied load. The microprestress relaxes in time and its evolution at each point of a concrete structure may be solved from the differential equation
where = positive constants. The microprestress can model the fact that drying and cooling, as well as wetting and heating, accelerate creep. The fact that changes of or produce new microprestress peaks and thus activate new creep sites explains the drying creep effect. A part of this effect, however, is caused by the fact that microcracking in a companion load-free specimen renders its overall shrinkage smaller than the shrinkage in an uncracked specimen, thus increasing the difference between the two.
The concept of microprestress is also needed to explain the stiffening due
to aging. One physical cause of aging is that the hydration products
gradually fill the pores of hardened cement paste, as reflected in function
in Eq.. But hydration ceases after about one year, yet the effect of the age at loading is strong even after many years. The explanation is that the microstress peaks relax with age, which reduces the number of creep sites and thus the rate of bond breakages.
At variable environment, time in Eq. must be replaced by
equivalent hydration time where = decreasing function of and . In Eq., must be replaced by where = reduced time, capturing the effect of and on creep viscosity; = function of decreasing from 1 at to 0 at ;, 5000 K.
The evolution of humidity profiles may be approximately considered as uncoupled from the stress and deformation problem and may be solved numerically from the diffusion equation divgrad

Approximate cross-section response at drying

Although multidimensional finite element calculations of creep and moisture
diffusion are nowadays feasible, simplified one-dimensional analysis of
concrete beams or girders based on the assumption of planar cross sections
remaining planar still reigns in practice. Although it involves deflection errors of the order of 30%. In that approach, one needs as
input the average cross-sectional compliance function and average shrinkage function of the cross section . Compared to the point-wise [constitutive equation, the algebraic expressions for such average characteristics are considerably more complicated and their accuracy is lower, especially if the cross section is not under centric compression. The following approximations have been derived and their coefficients optimized by fitting a large laboratory database for environmental humidities below 98%:
where = effective thickness, = volume-to-surface ratio, = 1 for normal cement; = shape factor ; and, = constant; . Eqs. and apply except that must be replaced by
where and. The form of the expression for shrinkage halftime is based on the diffusion theory. Function 'tanh' in Eq. 8 is the simplest function satisfying two asymptotic conditions ensuing from the diffusion theory: 1) for short times, and 2) the final shrinkage must be approached exponentially. Generalizations for the temperature effect exist, too.
Empirical formulae have been developed for predicting the parameter values
in the foregoing equations on the basis of concrete strength and some
parameters of the concrete mix. However, they are very crude, leading to prediction errors with the coefficients of variation of about 23% for creep and 34% for drying
shrinkage. These high uncertainties can be drastically reduced by updating
certain coefficients of the formulae according to short-time creep and
shrinkage tests of the given concrete. For shrinkage, however, the weight
loss of the drying test specimens must also be measured. A fully rational prediction of concrete creep and shrinkage properties from its composition is a formidable problem, far from resolved satisfactorily.

Engineering applications

The foregoing form of functions and has been used in the design of structures of high creep sensitivity. Other forms have been introduced into the design codes and standard recommendations of engineering societies. They are simpler though less realistic, especially for multi-decade creep.
Creep and shrinkage can cause a major loss of prestress.
Underestimation of multi-decade creep has caused excessive deflections, often with cracking, in many of large-span prestressed segmentally erected box girder bridges. Creep may cause excessive stress and cracking in cable-stayed or arch
bridges, and roof shells. Nonuniformity of creep and shrinkage, caused by
differences in the histories of pore humidity and temperature, age and
concrete type in various parts of a structures may lead to cracking. So may
interactions with masonry or with steel parts, as in cable-stayed bridges
and composite steel-concrete girders. Differences in column shortenings are
of particular concern for very tall buildings. In slender structures, creep
may cause collapse due to long-time instability.
The creep effects are particularly important for prestressed concrete
structures, and are
paramount in safety analysis of nuclear reactor containments and vessels.
At high temperature exposure, as in fire or postulated nuclear reactor
accidents, creep is very large and plays a major role.
In preliminary design of structures, simplified calculations may
conveniently use the dimensionless creep coefficient =. The change of structure state from time of initial loading to time can simply, though crudely, be estimated by quasi-elastic analysis in which Young's modulus is replaced by the so-called age-adjusted effective modulus.
The best approach to computer creep analysis of sensitive structures is to convert
the creep law to an incremental elastic stress–strain relation with an eigenstrain. Eq. can be used but in that form the variations
of humidity and temperature with time cannot be introduced and the need to
store the entire stress history for each finite element is cumbersome. It
is better to convert Eq. to a set of differential equations
based on the Kelvin chain rheologic model. To this end, the creep
properties in each sufficiently small time step may be considered as
non-aging, in which case a continuous spectrum of retardation moduli of
Kelvin chain may be obtained from by Widder's explicit formula for approximate Laplace transform inversion. The moduli of the Kelvin units then follow by discretizing this spectrum. They are different for each integration point of each finite element in each time step. This way the creep analysis problem gets converted to a series of elastic structural analyses, each of which can be run on a commercial finite element program. For an example see the last reference below.

Selected bibliography