Dinámica de fluidos computacional

There are two types of shrinkage that need consideration in design. Autogenous shrinkage occurs during the early months after setting of the concrete. It happens in practice in the interior of a concrete mass. The contraction of the cement paste is restrained by the rigid skeleton of the already hydrated cement paste and by the aggregate particles. According to BS EN 1992-1-1 autogenous shrinkage occurs in all concretes and is directly related to the concrete strength.

Drying shrinkage is the result of volume change that accompanies the loss of moisture over months and years from the concrete to the atmosphere. Thus the relative humidity (RH) of the environment, the amount and nature of the cementitious binder (cement Class) and the member size play important roles. Portland cement rich concretes will shrink more than leaner concretes. All other things being equal, shrinkage will decrease as cement Class changes from R to N to S. In the UK, indoor concrete will shrink more readily compared with external concrete, as the atmosphere will be drier. Similarly, as moisture loss will be easier in thin sections compared with thick members they will shrink more quickly.

Shrinkage strains develop over a long period (decades: See BS EN 1991-1-1 Exp. (3.9)).

Figure 9.3a shows the relative magnitudes of drying and autogenous shrinkage for a 450 mm and a 250 mm ground slab. Figure 9.3b indicates the proportion of the fi nal shrinkage that occurs at different ages of concrete. It will be seen that only about 70% of the fi nal shrinkage will have occurred after 10 000 days. The fi gures illustrate the general trends discussed above. It will be seen that autogenous shrinkage strain is an order of magnitude smaller than the long-term shrinkage strains (see also Tables A9 and A10).

Figure 9.2 Early thermal effects – effect of different variables on temperature drop.

Drying shrinkage - 400 mm slab Drying shrinkage - 250 mm slab Autogenous shrinkage

0.00025

0.00020

0.00015

0.00010

0.00005

0

1 10 100 1000 10000

Age of concrete - days

Shrinkage strain

Drying shrinkage and autogenous shrinkage C30/37 concrete using Class S cement RH 60%

(indoors) slab exposed on one face to atmosphere

a) Drying shrinkage and autogenous shrinkage

b) Drying of shrinkage with age of concrete

Note

Based on BS EN 1992-1-1 for a 400 mm and a 250 mm thick ground slab, assuming C30/37 concrete, Class S cement and 60% RH, one face exposed to atmosphere

Ground floor slab 250 mm thick Ground floor slab 400 mm thick

Development of shrinkage - C30/37 concrete - Class S cement

Shrinkage strain (t) / shrinkage strain (final)

1 10 100 1000 10000

Age of concrete - days 0.8

0.6 0.5 0.4 0.3 0.2 0.1 0 0.7 Figure 9.3

Development of shrinkage with age.

For slabs cast on the ground, consideration should also be given to the possibilities of cracking caused by differential drying shrinkage between the surface exposed to the air and that in contact with the ground. The gradient at the strains implies a curvature (1/r) = ε outside -ε outside/h) and a moment of EI (1/r) to be accounted for.

Exposed concrete

Clause 7.4 of BS 8110-2^[58] notes: Concrete exposed to the outdoor climate in the UK will exhibit seasonal cyclic strains of 0.4 times the 30 year shrinkage superimposed on the average shrinkage strain; the maximum shrinkage will occur at the end of each summer.

No similar guidance exists in BS EN 1992-1-1. As the seasonal temperature effects are usually greater (and allowed for in design), the seasonal variation of shrinkage can generally be ignored^[59].

9.1.3 Restraints

Tensile forces and cracks will only develop if movement is restrained. See the schematic model for end restraint shown in Figure 9.4.

Restraint may be external or internal to the element. External restraint takes two forms: restraint at ends and restraint along one or more edges. The mechanism of crack formation in these two cases is assumed to be different and this is refl ected in the formulae used for the calculation of crack inducing strain.

Internal restraint will be signifi cant generally in thick sections (about 1 m or more), in which differential expansion causes internal restraint. Internal restraint is not considered further in this publication. Where thick slabs are used reference should be made to CIRIA C660^[18].

Original length Free contraction: R = 0.0

Restrained movement due to partial

Restrained strain: stresses induced

Fully restrained: R = 1.0

Restrained strain: large stresses induced Rigid

restraint

Note:

A restraint factor, R, is used in the calculation of crack inducing strain. Many charts (e.g. Figure L.1 in BS EN 1992-3 include the effect of creep: thus ‘fully restrained’ appears as a factor of 0.5.

restraint: R = 0.0 to 1.0

Figure 9.4 Effect of end restraint.

End restraint

End restraint typically occurs in:

 infill bays

 ground slab cast on piles

 large area ground slabs, restrained locally, e.g. by piles, columns or column foundations, or by a build up of friction

 suspended slab cast between rigid cores, walls or columns

 walls cast against secant, contiguous concrete or steel sheet piled walls

In the case of members restrained at ends, each crack occurs to its full potential width before a successive crack occurs (see Figure 9.5). In this case crack-inducing strain is specifi cally related to the strength of the concrete and the steel ratio.

Edge restraint

The case of edge restraint, for example adjacent slab pours or a wall poured onto an existing base as illustrated in Figure 9.6, differs signifi cantly from the condition of end restraint. The principal difference is that, along with the steel, the adjacent concrete also acts as a crack distributor. Formation of a crack in this case only infl uences the distribution of stresses locally and the crack width is a function of the restrained strain rather than the tensile capacity of the concrete.