CAPÍTULO 6: RESULTADOS
5. RESULTADOS DEL ANÁLISIS DEL VALOR TOLERANCIA EN LOS
5.1. El valor de la Tolerancia en determinados cuentos mediante las TV
=
wc
F w dw
G
0
) (
σ , (3.19)
and was introduced in this concept for ordinary concrete by Hillerborg, Modéer and Petersson, via the fictitious crack model [Hillerborg et al. 1976]. The principles of this model are given in § 4.2, but only the definition 3.19 is included here, as a means of distinguishing between the nature of fracture energy as defined in linear elastic fracture mechanics theories and the energy consumed for the opening of the bridged crack. The latter is actually dissipated throughout the crack-bridging actions, which for an FRC include debonding, friction, deformation and possible fibre failure, as explained in
§ 3.3.2.3 and § 3.3.2.4.
There is also a significant difference in the definition of the crack, which is, in fracture mechanics terms, a stress-free surface, while the crack in FRC materials transfers a considerable amount of stress, and the term “fictitious” is found more appropriate to designate such material discontinuity.
However, on the macroscopic (structure-relevant) scale, the fictitious crack is the representative crack in fibre-reinforced concretes, while the stress free-microcracks in the matrix are insignificant on this scale. This justifies the concept of fracture energy as defined in Equation 3.19.
The fracture energy values of typical UHPFRC are in a range higher that 10 000 J/m2, while for ordinary concrete, fracture energy is of the magnitude of 100 J/m2, Table 3.2. It should be noted that the significance of the total fracture energy, GF, of UHPFRC is less important than that of brittle materials or even ordinary concrete. The initial rate of change in fracture energy for crack growth becomes more significant, as explained in the case of stable crack propagation, § 4.3.5.3.
3.3.2.7 Constitutive and design material curves in tension Constitutive curves in recommendations for UHPFRC
Present recommendations for UHPFRC suggest the use of tests (tensile or bending) for characterization of tensile behaviour. Transfer factor is proposed in order to derivate intrinsic curves from different test procedures [SETRA, AFGC 2002]. Figure 3.18 a) shows a proposal for constitutive curves in tension according to French recommendations, while a simplified stress-crack opening relationship, suggested by Japanese recommendations, is shown in the same figure b).
a) b)
Figure 3.18: Characteristic tensile curves: a) stress-strain and crack opening according to [SETRA, AFGC 2002]; b) idealised stress-crack opening according to [JSCE 2006]
Design curves in recommendations for UHPFRC
In order to simplify computations, current recommendations propose the use of a continuous stress-strain curve including the softening part. It should be remembered that “a complete stress-stress-strain curve does not exist as a material property; in case where such curves are shown, they have to be referred to a certain length over which the additional deformation within the fracture zone has been averaged”, [Hillerborg 1991].
a) b)
Figure 3.19: Design curves: a) by [SETRA, AFGC 2002]; b) by [JSCE 2006]
French recommendations [SETRA, AFGC 2002] introduce a quantity termed the characteristic length, lc, for transforming σ(w) (Figure 3.18 a)) into the σ(ε) curve (Figure 3.19 a)). The characteristic length is dependent on sectional dimensions, and for rectangular and T cross-sections, a value lc=2/3 h is proposed by these regulations, with h being the depth of the section. The transformation is based on the relationship
c c
ct
c E
f l w
+ ⋅
= γ
ε (3.20)
where γc is the appropriate design safety factor.
In Japanese recommendations, the equivalent specific length, Leq, is used to transform the σ(w) (Figure 3.18 b)) into the σ(ε) curve (Figure 3.19 b)). This value is calculated using numerical simulations, and is reported as being dependent on section height and shape. The expression for Leq
is based on the assumption that the flexural strengths, obtained by the FEM analysis using σ(w), is equal to the flexural strength obtained from the section equilibrium using the σ(ε) curve. The following relationship is derived from this numerical analysis:
( )
⎟⎟⎠⎞
⎜⎜
⎝
⎛
⋅
− +
⋅
⋅
= 4
/ 6 05 . 1 1 1 8 . 0
ch
eq h h l
L (3.21)
where
2 ct
c F
ch f
E
l G ⋅
= (3.22)
For the sake of comparison with the lc, for a typical UHPFRC with 2 % Vf, the value of the equivalent specific length Leq is obtained in the range of 0.16 h 0.35 h for thin elements (10 200 mm), that is smaller value, leading to higher tensile deformations, in comparison to lc=2/3 h proposed by the French recommendations. For more thick elements the difference between Leq and lc
becomes less significant.
Further comments concerning the characteristic length used to transform the σ(w) into the σ(ε) can be found in Chapter 4.
3.3.2.8 Tensile behaviour of UHPFRC used in present study; proposed material curves For the UHPFRC used in this study (Table 3.3) the data are provided from direct tensile tests. Tests were performed at the EPFL Structural Concrete Laboratory. Two types of test specimens were considered: unnotched (Figure 3.20 a)) and notched (Figure 3.20 b)). More information about these tests can be found in [Jungwirth 2006]. Unnotched specimens show lower tensile strength, coinciding with the strength of the statistically least resistant section.
a)
0 10
0 10 20 30
ε [‰]
σ [MPa]
b)
0 10
0 1 2 3
Δ l [mm]
σ [MP
a ]
Figure 3.20: Measured response of UHPFRC specimen in uniaxial tension: a) average deformation over measurement base lm=100 mm of unnotched specimens (G14T1, G14T3);
b) elongation of measurement base lm = 100 mm of notched specimens (G15F1, G15F2, G15F ); more detailed data are presented in [Jungwirth 2006]
Stress- strain curve
Based on the experimental data (Figure 3.20) and in accordance with the previous analytical considerations (§ 3.3.2.4) the multi-microcracking takes place throughout the volume of the element up to a certain deformation. Microcracks are spaced closely together, with small openings, invisible to the naked eye. For a representative volume relevant for the structural scale, the integral of localised deformations can be considered as a state of uniform change in the material’s structure, thus giving a homogeneous continuous behaviour, as if no localisation took place. From the bulk strain and stress increases, the average uniaxial values can be obtained as follows:
AA ixxdA
xx = 1
∫
σ ,σ , (3.23)
dx L L ixx x
xx 1 ( )
∫
,= ε
ε , (3.24)
where x is direstion of tensile force, and A is cross-section of the element.
For the studied UHPFRC, the tensile behaviour before localisation of deformations is represented by a bilinear σ(ε) curve (Figure 3.21). This curve is obtained respecting the integral of the force, based on the following assumptions:
- the elastic part is characterised by fct = 9 MPa and Ec = 60 GPa, (Appendix T1),
- multi-microcracking is represented by a hardening with zero slope, designated the pseudo-plastic plateau
- the pseudo-plastic plateau is limited to εu = 2.5 ‰, also according to [Jungwirth 2006].
The stress-strain relationship in unloading, Ed, is considered to be constant for a pseudo-plastic deformation attained (Figure 3.21 b)). For the tested material, the value of the unloading slope for the
maximal pseudo-plastic deformation attained, εu, is E* = 5.8 GPa, based on data from [Jungwirth 2006].
The advantage of a bilinear material model with zero hardening slope is that it enables the simple development of analytical expressions for prediction of element resistances. Moreover, it is shown in Chapter 4 that this model is appropriate for simulation of the behaviour of elements made of other UHPFRCs, and materials that exhibit slight strain hardening. Significant pseudo strain hardening slopes are very rarely observed in UHPFRC.
a) b)
0 1 2 3
Ε 11000
0 5 10
ΣMPa
σ
E
ct
εel εu
E ( )ε
f
c E*
d
Figure 3.21: a) Measured and assumed constitutive stress-strain relationship (up to ultimate homogenous deformation) for BSI UHPFRC; b) characteristic points of the σ(ε) curve (notations)
Stress-crack opening
The stress–crack opening, σ(w), curve characterises the material at the fracture zone. The analytical expression for design relevant σ(w) relationship can be formulated providing that the portions of fracture energies required for small crack openings are maintained. This yields the shape of the curve with the initial slope corresponding to the slope of the measured curve. The importance of this criterion is explained in Chapter 4.
According to § 3.3.2.6, the energy required to open the crack of a unit area from crack opening level wi to wi+1, is
∫
+= Δ
1
) (
i
i i
w
w
F w dw
G σ . (3.25)
If an interpolation function is found over the measured data, {{σ, w}}, an approximate σ(w) function is sought, verifying the accuracy of the approximate function using the equality of the energy portions (Equation 3.25) as follows:
∫
∫
+1 interpolation( ) = i+1 approximate( )i i
i
w
w w
w
dw w dw
w σ
σ (3.26)
For further analytical implementation in design, the additional criterion was the simplicity of the form of the approximate function, and its integrability.
It is found that the stress-crack opening relationship is satisfactorily approximated either by the multi-linear curve (Figure 3.22 b), Equation 3.32), or the curve of the following analytical shape (Figure 3.22 a)):
p
where both the parameters, wn,w and p, are determined from data fitting. Good correlation with test results was found for parameter p=1.2. The parameter wn,w corresponds to the crack opening at which stress decreases to the value of 2-p fct. In the case of the measured curves, with p=1.2, wn,w yields wn,w = 1 mm.
This curve is similar to the curve proposed by the empirical model of Stang [Stang et al. 1995], [Li 1993Li et al. 1993]:
where p and w1/2 are also the parameters to be obtained from experimental data, and, similarly to the previous curve, w1/2 corresponds to the crack opening when stress decreases to 0.5 fct.
The advantage of the curve proposed in Equation 3.27 is that the initial slope is a determined value:
w while, for p ≠ 1, it is indeterminate for Equation 3.28. Both curves exhibit the similar inaccuracy for
stress at higher crack openings: neither the stress nor the slope of the curves at w = wcryields zero, Figure 3.22 a). However, this inaccuracy occurs for large crack openings (> 6 mm), which, as shown in Chapter 4, are of no practical interest in the major part of the analysis (bending, bending with N force).
Figure 3.22: Material constitutive law for stress-crack opening relationship and measured data:
a) curve resulting from Equation 3.27; b) multilinear curve.
An additional advantage of Expression 3.27 is that it is an integrable function for w >0:
1
For the development of analytical expressions for further elements analysis, it was found more suitable and equally accurate to use a multilinear curve to describe the stress-crack opening relationship1: where the inflection points, {{wi, σ(wi)}}, can be found using the curve proposed in Equation 3.27 and condition 3.26. For the first part of the linear curve, the equality of the slopes (Equation 3.29) can also be used as a criterion for definition of the parameters in Equation 3.32. For the given data, the first inflection point is set at w1 = 0.5 mm.
Furthermore, considering that for the most frequent design needs (e.g. bending resistance), only the small crack openings range is attained, the first linear part of the σ(w) is typically sufficient.
Consideration of some other σ(w) curves
The curve proposed by Behloul’s model [Behloul 1996] (Equation 3.15) with approximation of the function F(α, r) by a polynomial curve (1-r)γ yields the shape similar curve shape is proposed by Kosa and Naaman [Kosa, Naaman 1990] with γ = 3. The initial slope of this curve is influenced by the ratio between tensile strength and critical crack opening (wcr, which is typically assumed as lf /2), and the parameter γ :
This curve was found suitable for different materials [Marti et al. 1999], including ordinary concrete [Kenel 2002] where γis found by the least-squares fit, as a value ranging between 4 and 6. However the initial slope with the mentioned range of γ is found unsatisfactory for describing the behaviour of the tested UHPFRC (Figure 3.23 a)). The reason is the difference in ratio between fct and wcr, (as parameters influencing the initial slope, Equation 3.34). Consequently, higher γ values enable a
1 Bilinear curves are commonly used for modelling of softening behaviour of ordinary and SFRCs [RILEM 2002];
better fitting of the first part of the curve, but in this case the rest of the curve is not well followed (Figure 3.23 b)). Additionally, it should be stated that, using a standard procedure for fitting the parameters (e.g. least-squares fit) over the whole set of data, the obtained curves may respect the integral
F w
e
approximat w dw G
cr
∫
=0
) (
σ ,
but stress values in the initial part of the curve are overestimated, which can lead to unreliable predictions of the element’s resistances (Figure 3.23 b)).
a) b)
0 5 10
w mm
0 5 10
ΣMPa
0 5 10
w mm
0 5 10
ΣMPa
Figure 3.23: Modelling of tensile strain softening according to Equation 3.33: a) with wcr=lf / 2 and γ = 2, 3 and 4; b) with wcr=lf / 2 and parameter γ obtained by fitting for three sets of measured data (γ =4.2, 5.1 and 6.3)
Another family of curves is based on the shape proposed by Shah [Shah 1987] :
wp
q ct e f w)= ⋅ − ⋅
σ( (3.36)
with q and p being fitting parameters. A plot of such a shape against the measured data is shown in Figure 3.24. This expression enables material behaviour to be represented with sufficient precision.
It is found less attractive for analytical application in the design of elements however. In cases where numerical integration is applicable, this form of curve can be well applied.
0 5 10
w mm
0 5 10
ΣMPa
Figure 3.24: Fitted exponential curve (Equation 3.36) with obtained parameters q = 0.7 and p = 0.75