where t0) is the creep coefficient related to Ec , the tangent modulus, which may be taken as 1,05 Ecm as from Table [3.1-EC2]. Annex B of the Eurocode gives detailed information on the development of creep with time. Where great accuracy is not required, the value found from Figure 3.1 may be considered as the creep coefficient, provided that the concrete is not subjected to a compressive stress greater than 0,45fck(tc) at an age to.
The values given in Figure 3.1 are valid for ambient temperatures between -40°C and +40°C and a mean relative humidity between RH = 40% and RH = 100%. Moreover, graphs are in function of the concrete to, expressed in days, at the time of loading, of the notional size ho = 2Ac/u where Ac is the concrete cross-sectional area and u is the perimeter of that part which is exposed to drying. They are also in function of the concrete class (e.g. C30/37) and of the class (R, N, S) of cement used, as detailed at clause [3.1.2(6)-EC2].
When the compressive stress of concrete at an age to exceeds the value 0,45 fck(to) then the proportionality expressed by [(3.1)-EC2] does not subsist and creep non-linearity should be considered. In such cases the non-linear notional creep coefficient should be obtained as from the exponential expression [(3.7)-EC2].
The total shrinkage strain £cs is composed of two components:
£cd, the drying shrinkage strain, which develops slowly, since it is a function of the migration of the water through the hardened concrete and £ca, the autogenous shrinkage strain, which develops during hardening of the concrete: the major part of it therefore develops in the early days after casting. Autogenous shrinkage can be defined as "the macroscopic volume reduction of cementitious materials when cement hydrates after initial setting. Autogenous shrinkage does not include the volume change due to loss or ingress of substances, temperature variation, the application of an external force and restraint" [JCI, 1998]. Autogenous shrinkage specially has to be regarded for higher strength concrete's, since its value increases with decreasing water cement ratio. Autogenous shrinkage is negligible, in comparison to drying shrinkage, in concrete having a w/c ratio greater than 0.45, but it can represent 50% of the total shrinkage when w/c is 0.30. Its development in time is linked to the hardening process of the concrete. In high strength concrete there is a considerable strength development during the first days; therefore autogenous shrinkage specially has to be regarded in cases that imposed deformations can occur, such as in the case that new concrete is cast against old concrete. In Annex B of EC-2 the basic equations for both drying shrinkage and autogenous shrinkage are given. They are valid up to a concrete strength class C90.
3.1.5 Stress-strain relation for non-linear structural analysis
Drying shrinkage is essentially a function of the ambient humidity and of the notional size ho = 2Ac/u. Clause [3.i.4(6)-EC2] gives formulae and tabled values normally used. Further information is given in Annex B (part B2).
Shrinkage according to EN 1992-1-1 42,5R tcure=1d - RH=65% hn=150mm
Shrinkage according to EN 1992-1-1 42,5R tcure=1d - RH=65% hn=150mm
Figure 3.2. Development of shrinkage according to EC2 C3.1.5 Stress-strain relation for non-linear structural analysis
Clause [3.i.5-EC2] gives the stress-strain relation for non-linear structural analysis as described by [Fig. 3.2-EC2] and by the expression [(3.i4)-EC2].
In the ENV-i992-i-i the following relation has been used in order to describe the mean stress strain relation:
Figure 3.2. Development of shrinkage according to EC2 C3.1.5 Stress-strain relation for non-linear structural analysis
Clause [3.i.5-EC2] gives the stress-strain relation for non-linear structural analysis as described by [Fig. 3.2-EC2] and by the expression [(3.i4)-EC2].
In the ENV-i992-i-i the following relation has been used in order to describe the mean stress strain relation:
£ci%o = 0.0022 (strain at peak compressive stress) k = (i.i Ec) eci/fc
Ec denotes the mean value Ecm of the longitudinal modulus of deformation, where
In ENV 1992-1-1, however, only concrete strength classes up to C50/60 were considered.
High strength concrete is known to behave in a more brittle way and the formulation therefore cannot be extended to high strength concrete without modification.
Fig. 3.2 shows compressive stress-strain relations for concrete strength classes ranging from about C25 to C90
Figure 3.2 Stress-strain relation for concrete's different strength classes subjected to a constant strain rate (strain in horizontal axis in stress in vertical axis in MPa)
Figure 3.2 Stress-strain relation for concrete's different strength classes subjected to a constant strain rate (strain in horizontal axis in stress in vertical axis in MPa)
In [CEB, 1995], the following modifications have been proposed:
- Eq. (4) overestimates the E-modulus for HSC. An appropriate formulation for HSC is:
where the difference between mean and characteristic strength Af is 8 MPa. This equation is also a good approximation for the E-modulus of normal strength concrete and could therefore be attributed general validity. (It should be noted that the given values are mean values and that the real modulus of elasticity can considerably be influenced by a component like the aggregate. If the modulus of elasticity is important and results from similar types of concrete are not known, testing of the concrete considered is recommended).
- To determine the ascending branch, using Eq. 1, the constant value £ci = - 0.0022 should for HSC be replaced by
This value can as well be used for normal strength concrete
- For HSC (>C50) the descending branch should be formulated by
where m = £c / £c1 and m2 = (£c1 + £o) / £c1 where £c0 is a value to be taken from Table 3.3.
Table 3.3. The parameter t for HSC
where m = £c / £c1 and m2 = (£c1 + £o) / £c1 where £c0 is a value to be taken from Table 3.3.
Table 3.3. The parameter t for HSC
fck (MPa) |
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