Once the data had been collected and processed, a deeper discussion and analysis of the findings was needed in order to assess the size effect in the fracture mode I of PFRC.

#### 4.1. Size Effect and Strength-CMOD Curves

The curves of the average strength-CMOD curves for the three tested sizes, as shown in

Figure 7, reveal clear behaviour differences. The residual strength obtained by using Equation (1), which is directly derived from the classical theory of strength of materials, already considers the size of the specimen for the calculation of the stress in the lower fibre of the ligament section. Therefore, the curves shown in

Figure 9 permit a direct comparison that was not possible by only observing

Figure 7. This is of high relevance because it enables a comparison of residual strengths such as

f_{R1} and

f_{R3} which are the main values of strength used in structural design.

Starting with the strength at the limit of proportionality (

f_{LOP}), the curve of the small specimens presents a higher peak-stress level compared with the other sizes, as can be understood by observing

Figure 9. The strength values at the limit of proportionality were 5.86 MPa, 5.07 MPa and 3.85 MPa for, respectively, the small, medium and large specimens. Therefore, the small specimens experienced stresses 52.2% higher than the large size specimens (the scale effect). In such a sense, the expected following size effect classical law for concrete occurs: the larger the size, the lower are the strengths obtained. Since the limit of proportionality (

f_{LOP}) is controlled by the concrete matrix, not by the fibres, the size effect observed is in accordance with the results of previous published works for plain concrete, such as [

32].

The slopes of the first loading branch for each of the sizes were also evident, as shown in

Figure 9b. The highest slope obtained in the tests of the small specimens showed a stiffer behaviour than those obtained with the medium and larger specimens. By the end of the discharge branch, for a CMOD of 0.42 mm, the strengths show similar values and are in the same order as the sizes because the strength is assumed by the fibres. This behaviour reveals the ductile behaviour of a fibre-reinforced concrete structural member.

As suggested before, the strength values at the proportionality limit (

f_{LOP}) trend to decrease as a consequence of the specimen size increase and behave inversely to size. Contrarily, the rest of residual strengths behaved in the same direct sense as the fracture surface size, the number of fibres and the specimen size. In

Figure 10 these tendencies can be observed for the three sizes with respect to the proportionality limit (

f_{LOP}), minimum strength value at the end of the discharge branch (

f_{MIN}) and the strengths for the CMOD values of 0.5 mm and 2.5 mm (

f_{R1} and

f_{R3}) in relation to specimen depth

D. It was then necessary to analyse the effect of the fibre counting and orientation on the residual strength in order to decouple these effects from the size effect.

For the evaluation of the size effect on PFRC, the analysis of the residual strength is a key point. In

Figure 11 the tendency lines of the residual strength values in relation to the number of fibres encountered in the fracture surface are shown. By observing

Figure 11, a clear tendency of steep slope in the small specimens and a smoother slope in the medium and large specimens can be stated. It is worth mentioning that as only one fibre dosage was used, the number of fibres remained for each size in close values.

In such a way, starting at the point of minimum residual strength beyond the first discharge branch (f_{MIN}) the curve shows increasing strength values which mean an increase in the fracture energy up to a second relative maximum (f_{REM}) and a smoothly decreasing residual strength. This behaviour shows the superior ductility and toughness of PFRC compared with unreinforced concrete.

The analysis of the relationship of residual strengths versus number of fibres also confirms the existence of the size effect.

Figure 12 was prepared to analyse the size effect by using the plot residual strength versus number of fibres which shows the superior behaviour in terms of strength of the small specimens compared with the larger sizes. Considering all the sizes and fitting the lines passing through the origin, the tendency of each size could be found. If the figure is observed in detail, for a specific number of fibres in the fracture surface the strength obtained follows the size effect expected tendency. That is to say, the larger the size the lower is the strength for the same number of fibres. In such a way, the size effect existence was observed by decoupling the influence of the fibre distribution on the fracture surface.

Figure 12 shows the values of

f_{R1}, f_{R3}, f_{min} and

f_{REM} as a function of the number of fibres in the fracture surface. As can be seen, keeping constant the volume fraction, the number of fibres in the fracture surface in the smallest specimens is about 50, in the medium-sized specimens around 100 and close to 200 for the largest specimens. To assess the size effect in

f_{min},

Figure 12 was performed. In this figure, the results of

Figure 11a were extended by a linear fitting for the comparison with the same number of fibres. It has been shown [

33] that residual strength in fibre-reinforced concrete is directly related with the number of fibres in the fracture surface.

Figure 12 shows that with the same number of fibres there is a clear size effect. Considering, for example, 50 fibres the strength of the large specimens (point A) corresponds to a minimum value of 0.48 MPa, whereas this value increases to 0.78 MPa (point B) for medium-size specimens and reaches 1.11 MPa (point C) in case of small specimens. This data means a 231% strength increment in the small specimens in relation to the big specimens. Again, this behaviour confirmed that the size effect occurred for PFRC elements, according to the classical theory of size effect for quasi-brittle materials [

32].

With this procedure it was possible to conclude that the size effect took place not only in the strength values depending on the concrete matrix (f_{LOP}) but also in the post-cracking strength values.