Eight SFRC and two plain mixes were cast including both RCC and CC. The aggregates used for CC mixes were river aggregates, whilst graded crushed granite (typical of the UK practice) was used for the RCC mixes to increase the bond between the paste and aggregate and to provide stability during rolling. The nominal maximum size of coarse aggregates was limited to 14 mm for RCC and to 10 mm for CC; the gradation of the aggregates is given in Jafarifar et al. (14). The mix proportions used, typical of UK practice, is shown in Table 1. A sulfo-aluminate low energy cement was adopted to reduce environmental impact. Recommendations of the BS 8500-1 (17) were used for CC to develop a designated mix that can be used for heavy-duty external paving. Considering cyclic wet and dry conditions with an intended working life of at least 50 years, a cement content of 380 kg/m3 and a water-cement ratio of 0.35 were required.

For the RCC mix design, the cement content of 300 kg was needed to achieve the same strength as for CC. The water content for RCC was chosen to yield the maximum dry density of the compacted mix; 7% for the used cement and aggregates. For CC mixes, the target slump was 70 mm.

Three different fibre types were used: one RF, and two IF. The recycled fibres from tyres used in this study had diameters in the range of 0.1 to 0.23 mm and a tensile strength of around 2000 MPa. These fibres have irregular shapes and lengths, so their length distribution is determined using optical measurements (8). In this study, 85% of the fibres had length in the range of 10 to 25 mm, and 50% of them in the range of 15–25 mm. Details of statistical length distribution can be found in Jafarifar et al. (14). Ecolanes (11) determined the optimum practical amount of recycled fibres in the range of 50–60 kg per cubic meter of concrete (or around 2–2.5% by mass). However, higher percentage of recycled fibres (4% and 6% by mass) and plain concrete mixes were also examined. For IF, 2% of fibres by mass of concrete is commonly used for suspended slabs. Two types of industrial fibres were examined: twincone (diameter 1mm, length 54 mm) and hooked-end (diameter 1mm, length 50 mm), all with the same nominal tensile strength (1100 MPa).

To compact RCC specimens in the laboratory, a purpose made apparatus having a fixed weight and a 1.6 kW electric vibrating hammer (16 to 32.5 Hz) was used (8). RCC specimens were cast in three layers (five layers for cylinders) each compacted for 60 seconds.

Flexural prisms are commonly tested under three or four point loading. For SFRC, the difference between the two tests (three or four-point bending test) is not so large when a careful non-linear computation is carried out (18). RILEM proposes a three-point bending test, but in this study a four-point load arrangement was adopted as it creates a region of constant moment at the middle of the prism and minimises the load-spreading effect at the point of load application (19).

Most standards (3, 20–22) require a notch at the middle of the prism in order to concentrate cracking at a predefined section, whereas some others (23, 24) do not require a notch. In an un-notched prism, cracking may form at any section in the middle one-third of the length, where the moment is maximum and constant. When the test aims to study fracture energy of concrete through load versus crack mouth opening, it is more appropriate to follow the notched-prism testing procedure. To obtain the bending moment versus curvature (or to obtain σ-ε curve), un-notched prism testing can also be used. In this study both notched and un-notched prism tests were undertaken for comparison purposes.

Three 150 × 150 × 550 mm prisms (per mix) were cast in steel-plate moulds, following the recommendations of BS EN 14651 (20). Notches were cut (25 mm height and 5 mm width) on the tensile side of the specimens at mid-span, using a rotating diamond blade. For RCC specimens the notch was perpendicular to the compacted layers and for CC it was on a side at 90º to the casting direction.

The specimens were tested under displacement control in a 1000 kN servo-hydraulic machine. To exclude torsional effects and to measure the net deflection at the mid-span (in relation to the neutral axis of the beam at its support), a yoke was used as specified by JSCE-SF4 (21). For the notched prisms, the rate of displacement was controlled by the crack mouth opening, while for the un-notched prisms it was controlled by the mid-span deflection using two transducers fixed to the yoke on both sides.

The mean compressive strength of the mixes was obtained from three cylinders 150 × 300 mm (per mix) based on BS EN 12390-3 (25). The specimen codes and a summary of characteristics of the mixes and specimens are given in Table 2, including the age of testing, mean compressive strength, mean bending E modulus, and the standard deviations. The mean bending elastic modulus has been obtained from bending test results from three specimens per mix and based on the formula derived by Alexander (26). Mid-span beam deflections versus applied loads are presented and discussed in Section 5.

Sectional inverse analysis is an iterative procedure to calculate the stress distribution through an individual cross-section. The key assumption is that plane sections remain plane. The load is increased gradually; the neutral axis depth and the stress distribution through the mid span are calculated at each step based on force equilibrium (4). The post-cracking σ-ε relationship is incrementally determined; for that, the moment versus curvature of the section is transformed into load versus mid span deflection (27). The path of the σ-ε curve is iteratively chosen so that the numerical load versus deflection (P−δ ) curve matches the experimental one and the error function is minimised.

The FE method can also be used for inverse analysis. Cracking can be described in two ways; discrete cracking and smeared cracking. The discrete crack model uses stress-crack opening (σ-w) relationship as the softening law. The crack(s) location is predefined using cohesive elements. This approach is computationally expensive (4) and is not useful when crack location(s) may vary. In the other approach, it is assumed that cracking is smeared over a characteristic length and the stress-strain (σ-ε) relationship is used to describe the softening law. The characteristic length in an FE model depends on element geometry and formulation (e.g. equal to the average element length for a first-order element). In this paper, the smeared crack approach is used to study the softening behaviour of SFRC mixes following a mesh sensitivity analysis.

Mesh sensitivity results are presented for a typical RSFRC and a plain mix, for three selected element sizes (25 mm, 37.5 mm, and 75 mm). The tensile multilinear s-erelationships used in the analysis were back-calculated from section analysis to obtain the best fit to the experimental average P−δ curves; these are shown in Figure 3. The P−δ curves obtained by the FE method and section analysis are shown in Figure 4 and Figure 5 for plain mixes and RSFRC, respectively.

As shown for plain concrete in Figure 4(a) and 4(b), there is clear evidence of mesh sensitivity for the element size bigger that 37.5 mm for both FE approaches. The results obtained for 37.5 mm and 25 mm element sizes match well and this means that refinement to smaller than 37.5 mm does not lead to a narrower crack band. Thus, it can be concluded that a crack band width bigger than 75 mm (2 × 37.5 mm) is formed in the middle as the boundary of the strain-softening region (or the fracture process zone). For the 75 mm element size, a softer behaviour is predicted, as the fracture energy is distributed further along the element resulting in a crack band-width wider than in the experiments.

For the RSFRC mixes, as shown in Figure 5(a) and 5(b), the results obtained for 75 mm element size are just slightly softer than the other sizes, showing that mesh refinement cannot considerably reduce the crack band width to less than 150 mm (2 × 75 mm). Hence, for the RSFRC, the fracture process zone covers a high percentage of the constant-moment zone (≈150 mm).

This mesh sensitivity analysis shows that 37.5 mm first order elements are appropriate for the examined mixes. The obtained widths of the fracture process zone for these mixes (> 75 mm for plain concrete and ≈150 mm for RSFRC) also comply well with the range predicted by Bazant and Oh (30) (2–6 times the average element dimension or the characteristic length), for a full development of the fracture process zone independent of the size effect. The size effect may arise from heterogeneity of concrete containing aggregates which are not necessarily small compared to the structural dimensions. Bazant an Oh (30) found that, for the optimum results, the ratio of the characteristic length to the aggregate size ranges from 1.5 to 4, and suggested limiting the minimum value of this ratio to around 3. This ratio is also respected by choosing 37.5 mm element size in this study (the average element length/ the maximum aggregate size ≈3).

As shown in Figure 4(c) and Figure 5(c), since the σ-ε relationship was determined from section analysis, as expected, the predictions shown for section analysis match extensively well the experimental results. Reasonable agreement is also reached between the three numerical approaches (section analysis, CSC and CDP), when an element size comparable to the fracture process zone is selected in the FE models (37.5 mm or smaller).

To examine its objectivity, the σ-ε obtained for the un-notched prisms is used for section analysis of the notched prisms. The experimental P−δ curves for the notched and un-notched prisms are shown in Figure 6, for two typical SFRC mixes of RCC and CC. It is seen that the existence of a notch significantly decreases the peak load, due to the reduced section at the mid-span.

The reduced height (i.e. the height at the mid-span) is applied in the equations given by Casanova and Rossi (27) to calculate load versus deflection (P – δ ) for the moment-curvature obtained from section analysis. Using the σ-ε obtained for the un-notched prisms, the predicted P – δ curve fits the experimental results of the notched prisms, as shown Figure 7(a) and 7(b) for RCC and CC mixes, respectively. This shows: 1) the stiffness of the notched prism can be approximated satisfactorily equal to the stiffness of an un-notched one with the same height as its mid-span; 2) the location and distribution of cracks through the constant-moment zone of the un-notched prisms did not distinguishably affect the results.

To assess the accuracy of the empirical methods given by RILEM TC 162-TDF (3) and Barros et al. (5), numerical sectional inverse analysis is used for a variety of SFRC prisms (incorporating IF and RF) given in Table 2. The obtained σ-ε results from the empirical methods are compared with the predictions obtained from the numerical inverse analysis. Using section analysis, the predictions of the empirical methods are compared with the experimental results to evaluate the effect of any σ-ε inaccuracies on the predicted energy absorption. Energy absorption is represented by the area under the P – δ curves by up to 3.0 mm mid-span deflection. The results for one SFRC mix (CC-C380-IT2-28d-N) and one RSFRC mix (RCC-C300-R2-90d-N) are presented in Figures 8 and 9, respectively. In these figures the σ-ε predictions, the tensile strengths and the regenerated P – δ curves are shown for the existing empirical methods as well as for the numerical inverse analysis. From these figures the following data can be obtained:

These data have been obtained for all the examined mixes and are compared in Table 4.

The comparison shows that:

For the examined mixes in this study, the following equations have been statistically calibrated to the experimental load-deflection results to directly determine the multilinear s–e relationship. The following equations can reasonably cover the behaviour of CC and RCC reinforced with IF or RF [3].

Where, residual flexural tensile strength parameters, f_Rj (j = 1, 2, 3, 4), are defined as a fictitious maximum tensile stress with linear stress distribution in an uncracked mid-span section of a prism subjected to a load P_j corresponding to deflection δ_j (δ ₁ = 0.5 mm, δ ₂ = 1.3 mm, δ ₃ = 2.2 mm, δ ₄ = 3.0 mm); f_Rj = P_jl/bh ². Linear extrapolation is applied to strain values bigger than 20 ‰.

Using this multilinear σ-ε relationship the energy absorption is only overestimated by 2% (SD 6%) on average; the predicted tensile strength and total strain in the softening regime are also predicted extremely well (see Table 4).

It should be noted that the proposed relationship is based on experiments conducted on small-scale specimens. For real-scale structures, it is recommended to examine satisfactory reproduction of the results by conducting large-scale bending tests.