SANCIONADOR COMO INSTRUMENTO DE PROTECCIÓN AMBIENTAL
A. Distribución de competencias entre el Estado y las Comunidades Autónomas
Basically, the fracture energy, GF, is calculated from the area under the splitting force versus CMOD curve divided by the fracture surface. The fracture surface in the WST specimen is obtained by lligament times Width of the specimen; see Figure 5.4(a). The input energy could be obtained from the area under the vertical force versus actuator displacement, u-curve. From theoretical point of view, the input energy and the fracture energy should be the same, but larger energy is determined from the vertical load vs. vertical displacement curve. Because the vertical displacement includes not only the specimen displacement but also testing machine displacement, thus the energy calculated based on the total vertical displacement is larger than the fracture energy. It should be noted that, in the WST, the effect of self-weight is negligible even for large specimens; this is an important advantage compared to the Three-point Bending Test (3PBT), where the fracture energy due to the self-weight of the beam reaches up to 40 – 60 % of the total fracture energy; see Shah and Carpinteri (1991).
The contribution of the vertical force, Fv, in the fracture energy should also be taken into consideration. During the loading, when the specimen opens, the place of resultant undergoes some displacement in the vertical direction as well; see Figure 5.4(b). The energy, dissipated in the vertical direction, can be estimated from the area under the Fv versus vertical displacement curve, which is obtained by comparing the geometrical situation of the untracked (zero crack opening displacement) with the cracked (crack opening displacement of some finite value) specimen. A portion of 5%
to 9% of the fracture energy, obtained under the Fs – CMOD curve, was estimated for the cubical and cylindrical specimen by Shah and Carpinteri (1991). Based on the analysis, done in the current project, this value was approximated about 7.5% for cubic specimen of 150x150x150 mm3; for the detail calculation see Chapter 6. The more the load point is away from the axis of symmetry, the more important is this contribution (cylindrical specimen), and a very long WST specimen would correspond to the 3PBT beam; see Figure 5.4(c).
lligament
width
fracture surface
(a)
Fsp Fsp
Fv/2 Fv/2 Fv
(b) (c)
Figure 5.4 WST specimen, (a) Fracture surface, (b) Vertical and horizontal component of force, (c) Wedge splitting specimen as a ‘’compact’’
three-point-bending test; from Shah and Carpinteri (1991)
In order to determine -w relationship, the approach which has been developed by Löfgren (2005) can be used. The approach has been developed for FRC and includes three steps: (1) the material testing, e.g. the WST or the 3PBT; (2) inverse analysis (using non-linear fracture mechanics) where the -w relationship is determined; and (3) adjustment of the -w relationship for any differences in fiber efficiency (the number of fiber) between the experiment and random 3-D orientation or the member where the material is to be used. The same approach, but without the inverse analysis step, may be used for the uni-axial tension test.
Material Testing
The general requirements that has been stated by Löfgren (2005) for the first step, i.e.
material testing, are as follows:
• The material testing must provide results which readily can be interpreted as constitutive material parameters (the -w relationship).
• It should, preferably, provide a relationship between load and crack opening (or CMOD) which can be used for inverse analysis.
• The specimen should be designed such that a single, well-defined crack is formed, which generally means that the specimen has to be equipped with a notch of sufficient dimension.
• It should give representative values;
• It should, if possible, not require an advanced testing equipment or demand a high machine stiffness.
• It should be easy to handle and execute.
Inverse Analysis
The -w relationship can be determined by bi-linear, exponential or poly-linear (multi-linear) approximation; see Figure 5.5. Inverse analysis is achieved by minimising the difference between calculated displacement and target displacement, e.g. CMOD, obtained from test results; see Figure 5.6. The results taken from the 3PBT and the WST, can be analysed by inverse analysis in order to determine the -w; for more comprehensive review of inverse analysis the reference is made to Löfgren (2005).
Figure 5.5 Different -w relationship: (a) bi-linear; (b) exponential; and (c) poly-linear (or multi-poly-linear); from Löfgren (2005).
In the present project, the bi-linear approximation of the -w relationship is investigated by inverse analysis for both plain and fiber reinforce concrete. The recommendations, proposed by Löfgren (2005), has been taken into consideration when the inverse analysis was performed on FRC; see Chapter 6.
Figure 5.6 Principle of inverse analysis; from Löfgren (2005)
Adjustment of the -w relationship for fiber efficiency
The last step is to adjust the -w relationship for specific fiber efficiency by considering the actual number of fibers crossing the fracture surface. Löfgren (2005) has shown that the specimen size, in relation to the fiber length, has a considerable impact on the fiber efficiency factor. He has also shown that if the material test specimen, used for material characterisation, has a more or less random 3-D fiber orientation but with a different fiber efficiency factor compared to random 3-D; an experimentally determined fiber efficiency factor, b.exp, can be used to modify the stress-crack opening relationship so that it more closely corresponds to that of a completely random 3-D orientation. A linear relationship between the number of
fibers and the fiber bridging stress exists and it is possible to adjust the -w relationship obtained from inverse analysis, b.exp(w), considering the difference in fiber efficiency factor between material test specimen and the theoretical value for random 3-D orientation, b.3-D, according to:
b.exp is the experimentally determined fiber efficiency factor, and
b.3-D is the theoretical fiber efficiency factor for the 3-D case, which is equal to 0.5; see Löfgren (2004).
The Equation ( 5.3) provides the -w relationship for random 3-D orientation, b.3-D(w). The experimental fiber efficiency factor, b.exp, for the material test specimen can be determined by counting the number of fibers crossing the fracture plane and calculating with the following experiences:
f