Design-oriented approach to determine FRC constitutive law parameters considering the size effect
Introduction
Size effect has been an issue extensively studied and reported in the literature, with different theoretical approaches arising to explain such effect. According to previous investigations, using specimens of different dimensions to determine the flexural response of fibre reinforced concrete (FRC) has several implications regarding size effect that may influence the nominal strength. While some studies [1] conclude that the size effect on the flexural strength is almost negligible, other researchers [2] state that the size effect cannot be disregarded. This issue was addressed in additional investigations [3] by analysing the residual strength of different sized concrete specimens with and without fibres and concluded that increasing the size of the specimen leads to a reduction in the strength [4].
Despite the evident presence of size effect on concrete, most of the design codes and standards currently into effect still assume that the behaviour of concrete follows the classical theories of elasticity and plasticity [5]. In this regard, it is broadly accepted that both tensile and flexural strength capacities of concrete are not affected by the size effect at the structural design level. In the case of FRC, the fact of being a relatively new material for design purposes has also led to generally assume there is no size effect.
Among the existing codes and guidelines with specific FRC constitutive laws, only the German code (DBV) and the RILEM recommendations [6] account for the size effect by introducing a correction factor to reduce the strength as the size of the specimen increases [7]. Unlike the DBV and the RILEM specifications, the constitutive model for FRC of the fib Model Code 2010 (MC2010) [8] assumes an equivalent residual strength between the standard beam and the structural element. In this line, it has been reported [9] that the direct application of constitutive models on real-scale elements without considering the size effect may lead to an unsafe design given the influence of the geometry differences and the variations of the fibre distribution and orientation depending on the size of the element. For this, the MC2010 suggests considering an orientation factor to take into account the favourable or unfavourable effect of the fibres [10], [11].
Constitutive models for the design of FRC real-scale elements are usually based on the results of standard prismatic beams tested under a three-point bending configuration. In this regard, this study aims to analyse whether small specimens may be used to determine the post-cracking parameters of FRC and which assumptions need to be taken for this purpose of guaranteeing the required reliability of the resulting design parameters. Hence, the cracking mechanism is analysed in detail to propose an approach based on replacing the crack mouth opening displacement (CMOD) for the rotation of the specimen as the reference parameter to determine the residual strength of FRC. The method consists of an inverse analysis to analytically compute the FRC flexural performance, which allows obtaining the load-deflection or load-crack opening curves by using the constitutive law of the MC2010. The analytical results are consequently compared to experimental results conducted on prismatic beams of different sizes (40 × 40 × 160, 100 × 100 × 400 and 150 × 150 × 600 mm) to determine which reference parameter—CMOD or rotation—presents the closest results to the experimental curves.
Finally, an alternative approach to assess the constitutive law for FRC using small non-standard specimens is proposed based on the obtained results. This approach can lead to simplify the testing procedure while providing more representative results for thin or slab-shaped FRC elements, which are commonly subjected to a greater influence of the preferential fibre orientation that takes place [12], [13].
Section snippets
Constitutive law for FRC
Different constitutive models in varying degrees of complexity and accuracy may be found in the literature and national or international codes for FRC [7] and even for ultra-high performance fibre reinforced concrete [14]. The main particularity of the stress-strain tensile law of the MC2010 for FRC with respect to other codes is that it can distinguish among three cases of softening and hardening behaviour (Fig. 1). The branches describing the strain-softening or strain-hardening post-cracking
Materials and concrete mixes
One plain concrete mix (M0) and four high-performance fibre reinforced concrete mixes with contents of and of steel straight microfibres (, and ) were produced (Table 1). As in other studies [29], the content of fibres was increased by replacing an equivalent volume of silica sand to keep constant the content of cement.
Nanosilica was introduced in all mixes as a highly reactive pozzolanic material in a content of 5% over the cement weight (o.c.w.) to
Compressive strength and modulus of elasticity
The average results of the compressive strength for three different sized cubic specimens and the results of the modulus of elasticity performed on standard cylindrical samples are detailed in Table 3.
A comparison between FRC mixes with different water-cement ratios reveals an expected reduction of the compressive strength for higher ratios. This effect may also be appreciated in the modulus of elasticity, with lower values for higher ratios. The higher content of water in the mix
Conclusions
This study confirms the influence of the dimension of the specimens on the flexural strength and proposes an alternative approach to determine the parameters of the constitutive law for FRC based on the specifications of the MC2010 to analyse and predict the behaviour of small elements. The conclusions drawn from this study state as follows:
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The results show that size effect is significantly influenced by the effect of the orientation and content of fibres. The specimens of mixes blended with
CRediT authorship contribution statement
Eduardo Galeote: Conceptualization, Methodology, Investigation, Writing - original draft. Ana Blanco: Conceptualization, Supervision, Writing - review & editing. Albert de la Fuente: Conceptualization, Supervision, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The first author acknowledges the Spanish Ministry of Science, Innovation and Universities for the FPU13/04864 grant. The authors also want to express their gratitude to the Spanish Ministry of Economy, Industry and Competitiveness for the financial support through the SAES project (BIA2016-78742-C2-1-R).
Data availability
The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.
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