3. JUSTIFICACIÓN
3.1. Sustentación
Khennane and Baker (1993) take as a starting point the model developed by Anderberg and Thelandersson. However, they determine the instan-taneous stress-related strain using a stress–strain curve which is initially linear to a yield value which may be taken as 0,45 times the peak concrete strength and then the remainder of the characteristic is taken as a part of a quarter ellipse (Fig. 5.34) with the following equation
σc,θ− σ1,cθ2
σ0,c,θ− σ1,c,θ
2 +
p− εp,c,θ
2
p2 = 1 (5.104)
The final stress–strain law is in the form of an incremental rule
εtot,c = Aσc− Bσc+ εtr,c+ εth,c (5.105)
s0,c,θ
s1,c,θ
e0,c,θ
ep,c,θ
∆E
∆p
Ellipse
Tangent point
Strain
Stress
Figure 5.34 Linear elastic-elliptical plastic idealization of the stress–strain curve for concrete (after Khennane and Baker, 1993).
where εtot,c is the increment in total strain, σc is the stress, σc is the increment in stress over the time step, εth,c is the increment of thermal strain, εtr,cis the increment in transient strain defined in the same man-ner as Anderberg and Thelandersson and A and B are parameters defined by the following equations
A= Et
(Et+ βEt)2 + Ht
(Ht+ βHt)2+ β k2 σ0,c,20
∂εth,c
∂θ θ (5.106)
and
B= Et
(Et+ βEt)2 + Ht
(Ht+ βHt)2 (5.107)
where θ is the temperature, Et is the slope of the linear portion of the stress–strain curve, i.e. the initial tangent modulus, Etis the change in tangent modulus at time t to t + t, and Ht and Ht are the values of the strain hardening parameter and the change in the strain-hardening parameter, β is an interpolation parameter taking a value between zero and unity and θ is the temperature rise. Khennane and Baker found that the best value for β was 0,5.
The equivalent plastic strain εp,c,θ (determined from Eq. (5.105)) and the strain-hardening parameter H are both dependant upon the current stress state. The latter parameter is given by
H=
σ0,c,θ− σ1,c,θ
2
p2 p− εp,c,θ
σc,θ− σ1,c,θ
(5.108)
This model does not appear to allow for the instantaneous strain in the concrete to exceed the peak value and the formulation for transient strain valid for temperatures above 550◦C also appears not to be considered.
This latter point may be critical since Khennane and Baker appear to pro-duce excellent correlation between experimental results and prediction below temperatures of 550◦C but far poorer correlation at temperatures above this value. Further, the value of Ht needs to be estimated as it depends on the stress at the end of the incremental time step. The authors also indicate that an elaborate algorithm is needed to allow for the situa-tion when both the temperature and the stress vary during an incremental step. It should be noted that this is likely case in a full structural analysis of fire affected concrete members.
5.3.2.5 Schneider
The model used by Schneider is based on a unit stress compliance func-tion, i.e. the creep is considered to be linear with respect to stress. The general background to this approach is detailed in Bažant (1988) and the specific formulation for high temperatures is given in Bažant (1983).
An important simplification to the general compliance function approach that can be made for creep in a fire is that the duration of between a half and four hours is short compared with the age of the concrete and thus any time dependence in the model can be ignored. The full background to Schneider’s model is given in Schneider (1986b, 1988), and only the results will be presented.
The unit stress compliance function J(θ,σ ) can be written as
J(θc, σc)= 1+ κ Ec,θ +
Ec,θ (5.109)
where Ec,θ is the temperature-dependant modulus of elasticity and κ is a parameter allowing for non-linear stress–strain behaviour for stresses above about half the concrete strength and is given by Eq. (5.110) which is derived from Popovics (1973) (Eq. (5.37)),
κ= 1 n− 1
εσ,c,θ ε0,c,θ
n
(5.110)
with n taking a value of 2,5 for lightweight concrete and 3,0 for normal-weight concrete. An alternative formulation for κ is given by
κ = 1 n− 1
σ (θc) σ0,θ
5
(5.111)
The value of Ec,θ is given by
Ec,θ = gEc,0 (5.112)
where the parameter g is given by the following equation
g= 1 + fc,θ σ0,c,20
θc− 20
100 (5.113)
where θc is the concrete temperature (◦C), fc,θ/σ0,c,20 is the ratio of the initial stress under which the concrete is heated to the ambient strength and the creep function is given by
= gφ + fc,θ σ0,c,20
θc− 20
100 (5.114)
with φ being given by
φ= C1tanh γw(θc− 20) + C2tanh γ0 θc− θg
+ C3 (5.115)
with γw defined by
γw= 0,001 (0,3w + 2,2) (5.116)
where w is the moisture content in per cent by weight.
It should be reiterated that the stress used in the definition of g and is that initially applied at the start of the heating period.
The values of C1, C2, C3, θgand γ0proposed by Schneider are given in Table 5.7. There is however some evidence that these parameters are likely to be functions of the concrete mix proportions since Purkiss and Bali (1988) report values of 2,1 and 0,7 for C2and C3. A further unpublished analysis by the author of Bali’s original data (Bali, 1984) gives slightly different values of 1,5 and 0,95, respectively. An analysis by the author of Anderberg and Thelandersson’s data gives values for C2 and C3 of 3,27 and 1,78, respectively, for the tests with a heating rate of 1◦C/min.
It should be noted that the formulation by Anderberg and Thelandersson, whilst not justifiable on theoretical grounds has been suc-cessfully used as a model in computer simulations as has the Schneider model.
Table 5.7 Concrete stress–strain model parameters
Concrete type Parameter
C1 C2 C3 γ0(◦C) θg(◦C)
Quartzite 2,60 1,40 1,40 0,0075 700
Limestone 2,60 2,40 2,40 0,0075 650
Lightweight 2,60 3,00 3,00 0,0075 600
Source: Schneider (1985) by permission
1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1
00 0,005 0,01 0,015 0,02 0,025 0,03
Strains (stress-induced + transient), e Stress, s/sua
Anderbergy and Thelandersson Li and Purkiss
EN 1992-1-2
Figure 5.35 Comparisons of full stress–strain curves at temperatures 40, 100, 200, 300, 400, 500, 600 and 700◦C.