Las dimensiones de la competencia cívica: comentarios generales 

A considerable amount of work has been carried out by different researchers (e.g. Lim et al., 1987, Lok and Pei, 1998; Tlemat et al. 2006) and various tensile stress-strain constitutive models were proposed. Some of these models were discussed in Section 2.3 of Chapter 2. Amongst all the models discussed, only four models are generic (i.e. allowing for different volume fractions, bond stress, aspect ratio etc) and are thus relevant to the present research work. These models were proposed by Lim et al. (1987), Murugappan et al. (1994), Lok and Pei (1998) and Lok and Xiao (1999). Generally, the model proposed by Murugappan et al. (1994) was a modification on the model proposed by Lim et al. (1987). As both models were designed for steel fibres in the range of 0.5% to 1.5%, these models provided insufficient information with regards to the fibres content range requirement intended for the parametric study in the present

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work. Furthermore, it should be noted that Lim et al. (1987) model ignored the residual stress developed in the concrete matrix.

As a consequence of the aforementioned issues, the SFRC models proposed by Lok and Pei (1998) and Lok and Xiao (1999) were considered for calibration work. In addition, the models proposed by Tlematb et al. (2006) and Barros and Figueiras (2001) were also considered. The inclusion of these two models and their respective experimental data was useful as one model (Tlematb et al., 2006) allowed for high values of fibre volume fraction (i.e. up to 6%), while the other (Barros and Figueiras, 2001) was applicable only to low fibre amounts (e.g. 0.5%). This wide range was used to examine the capability of the two main models considered (i.e. Lok and Pei, 1998 and Lok and Xiao, 1999) to simulate the behaviour of SFRC with fibres provided in both low and high amounts. This is important since the range considered in the present work was quite varied, with volume fractions between 0.5%~2.5%.

All the aforementioned four models (i.e. Lok and Pei, 1998, Lok and Xiao, 1999 Tlematb et al., 2006 and Barros and Figueiras, 2001) were implanted into the commercially available finite-element software package ABAQUS (2007) as part of the calibration work. The results of the latter were then used to select the best model to be adopted in the subsequent parametric studies. The basic characteristics of the material models and the assumptions made are discussed in the following sections. The details and results of the calibration work were presented in Appendix A.

3.2.2 Tension model

The structural response of SFRC structures is predominantly characterised by its tensile behaviour. Therefore, SFRC behaviour can be effectively modelled using a suitable tensile stress-strain relationship. As mentioned earlier, four constitutive models were considered for the preliminary calibration work (i.e. Lok and Pei, 1998, Lok and Xiao, 1999; Tlematb et al., 2006; and Barros and Figueiras, 2001).

The value for the bond stress (_d) for Lok and Pei (1998) and Lok and Xiao (1999) models was assumed based on values summarised in Table 2.2 (Chapter 2). The values of ultimate compressive and tensile strain at failure were taken as -0.0035 and 0.02 (Craig et al., 1987; Lok and Pei, 1998; Lok and Xiao, 1999; Eurocode 2, 1992),

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respectively. In addition, if the tensile strength value was not given, the tensile stress was assumed as maximum of 10% of the compressive strength of the concrete.

A minimal conservative modification was made to the Lok and Xiao (1999) model as part of the present research work. The vertical line at the end of the SFRC tensile stress- strain curve proposed by Lok and Xiao (1999) was altered so that the stress decreases gradually to 0 between strain values of 0.018 to 0.02 rather than the original sudden vertical drop (refer to Figure 2.16 in Chapter 2). This alteration was suggested to avoid numerical instability, without affecting the accuracy of the results as the change is insignificant in practical terms. Additionally, the values used to define the material models proposed by Barros and Figueiras (1999) and Tlematb et al. (2006) were taken from the Tables 2.3 and 2.6, respectively.

3.2.3 Compression model

It was concluded in Chapter 2 that the addition of steel fibres has no significant effect on the compressive behaviour of concrete (Bencardino et al., 2008). Therefore for the present work, the SFRC compressive behaviour is assumed to be the same as the one for plain concrete.

3.2.4 Conclusions on SFRC constitutive models

Lok and Pei (1998) and Lok and Xiao (1999) are alike and adopt the same concepts to define the characteristic points on the stress-strain diagram. Lok and Xiao (1999) model, is an improvement of the Lok and Pei (1998) model, considering a user defined value for the orientation factor. Moreover, the shape of the post-cracking tensile stress- strain diagram is similar to the one proposed by RILEM TC 162-TDF Recommendation (2000; 2003) and other researchers (Lim et al., 1987; Murugappan et al., 1994).

Based on the findings of the calibration work carried out (with full details presented in Appendix A), it was found that the results produced by the Lok and Xiao (1999) constitutive model were in best agreement with existing experimental data used in the calibration study. Even in instances where there was a slight difference, the discrepancy was always on the safe side (in design terms) with the model predictions not overestimating actual strength results.

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Therefore, the model proposed by Lok and Xiao (1999) was chosen to be used in the present work based on the following reasons. First, the model is applicable for a reasonably wide practical range of fibre volume fractions (i.e. 0.5% to 3.0%), which is similar to the range investigated in the present research work. Secondly, the model is versatile as it allows for definition of different values of aspect ratio (L/d) and bond stress (τd). Thirdly, the randomness distribution of the steel fibres is considered in the

model. Fourthly, the model is capable of exhibiting both tension softening and hardening depending on the amount of fibres provided. Finally, through the calibration work carried out, all predictions were found to agree well with experimental results and to be always on the safe side in terms of its load carrying capacity estimates.