Elsevier

Composite Structures

Volume 239, 1 May 2020, 112036
Composite Structures

Design-oriented approach to determine FRC constitutive law parameters considering the size effect

https://doi.org/10.1016/j.compstruct.2020.112036Get rights and content

Abstract

Tensile strength constitutive laws for fibre reinforced concrete (FRC) are commonly defined through the parameters of flexural tests conducted on standard prismatic specimens. However, there are no specific criteria to determine such parameters using small specimens that could simplify the testing procedure and provide more representative results of slender structural FRC elements. In this line, the influence of size effect becomes an issue particularly relevant during the characterisation stage given that the residual strength decreases while increasing the size of the element. The objective of this document is to propose a methodology to obtain the parameters of the constitutive law using small specimens. For this, FRC residual strength was determined through three-point bending tests on prismatic notched beams of 40 × 40 × 160, 100 × 100 × 400 and 150 × 150 × 600 mm. An analytical model based on sectional analyses aimed at reproducing the flexural strength of FRC was used to assess the results of the alternative methodology to determine the parameters for the constitutive law. The results show that an approach based on the rotation instead of the crack opening as the reference parameter to estimate the stresses for the constitutive law leads to results less influenced by the size effect when designing small elements.

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 K 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 90 and 190kg/m3 of steel straight microfibres (l=13mm, ϕ=0.20mm and fyu=2750MPa) 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 (w/c) reveals an expected reduction of the compressive strength for higher w/c ratios. This effect may also be appreciated in the modulus of elasticity, with lower values for higher w/c 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:

  • 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.

References (51)

  • A. de la Fuente et al.

    Numerical model for the analysis up to failure of precast concrete sections

    Comput Struct

    (2012)
  • L. Liao et al.

    Design procedure and experimental study on fibre reinforced concrete segmental rings for vertical shafts

    Mater Des

    (2016)
  • B. Mobasher et al.

    Backcalculation of residual tensile strength of regular and high performance fiber reinforced concrete from flexural tests

    Constr Build Mater

    (2014)
  • B. Mobasher et al.

    Analytical solutions for flexural design of hybrid steel fiber reinforced concrete beams

    Eng Struct

    (2015)
  • D.-Y. Yoo et al.

    Structural performance of ultra-high-performance concrete beams with different steel fibers

    Eng Struct

    (2015)
  • J.Á. López et al.

    A simplified five-point inverse analysis method to determine the tensile properties of UHPFRC from unnotched four-point bending tests

    Compos Part B Eng

    (2016)
  • K. Wille et al.

    Properties of strain hardening ultra high performance fiber reinforced concrete (UHP-FRC) under direct tensile loading

    Cem Concr Compos

    (2014)
  • E. Ghafari et al.

    The effect of nanosilica addition on flowability, strength and transport properties of ultra high performance concrete

    Mater Des

    (2014)
  • C.G. Hoover et al.

    Comprehensive concrete fracture tests: description and results

    Eng Fract Mech

    (2013)
  • N. Trivedi et al.

    Investigation on fracture parameters of concrete through optical crack profile and size effect studies

    Eng Fract Mech

    (2015)
  • K. Kirane et al.

    Size effect in Paris law and fatigue lifetimes for quasibrittle materials: modified theory, experiments and micro-modeling

    Int J Fatigue

    (2016)
  • M.N. Soutsos et al.

    Flexural performance of fibre reinforced concrete made with steel and synthetic fibres

    Constr Build Mater

    (2012)
  • D.-Y. Yoo et al.

    Material and bond properties of ultra high performance fiber reinforced concrete with micro steel fibers

    Compos Part B Eng

    (2014)
  • M. Pająk et al.

    Flexural behavior of self-compacting concrete reinforced with different types of steel fibers

    Constr Build Mater

    (2013)
  • D.-Y. Yoo et al.

    Effects of fiber shape, aspect ratio, and volume fraction on flexural behavior of ultra-high-performance fiber-reinforced cement composites

    Compos Struct

    (2017)
  • Cited by (8)

    • Numerical and analytical optimisation of functionally graded concrete incorporating steel fibres and recycled aggregate

      2022, Construction and Building Materials
      Citation Excerpt :

      Apart from that, due to its simplicity and accuracy, an analytical model based on an analysis of evolutionary sections (AES) [34,35] may be used to simulate the flexural behaviour of FGC instead of a FEM model. The AES model has been successfully used by several researchers to simulate different fibre reinforced concrete structures [34–38]. In this context, the main objective of this study is to demonstrate the feasibility of using FGC in construction projects by estimating the ratio of bottom layer to total thicknesses (h/H) that results in an optimal post-cracking flexural response of FGC beams produced with PCC and FRRAC.

    • Neural network-aided prediction of post-cracking tensile strength of fibre-reinforced concrete

      2021, Computers and Structures
      Citation Excerpt :

      Similarly to the previous case, no studies were found in the literature addressing this particular phenomenon for the Barcelona test. Previous studies conducted on beams under flexure have shown that the residual strength of FRC decreases with the increase of specimen dimension [92,96-98] as a result of the size effect in concrete. In FRC, the greater strength of smaller elements may also be attributed to an increasing number of fibres oriented in a perpendicular direction to the cracked surface when reducing the dimension of the element [83].

    • Prediction of the residual flexural strength of fiber reinforced concrete using artificial neural networks

      2021, Construction and Building Materials
      Citation Excerpt :

      Fig. 1 presents the general workflow for the methodology developed in this article. First, a database of 400 three-point bending test residual strength datasets (EN 14651) was compiled from different sources in literature [51-88]. Subsequently, an artificial neural network model was developed using the Deep Learning Optimization Toolbox in MATLAB 2021a ® and trained according to the Levenberg-Marquardt optimization method [89].

    View all citing articles on Scopus
    View full text