## 5 Results and Discussion
### Compressive-Splitting Tensile Strength Relationship  
Figures 12 and 13 illustrate the correlation between the splitting tensile and compressive strengths of HSC with various contents of SF, PPF, NS and SP. Besides, formulas are proposed to consider the effect of SF and PPF. The proposed relationships (with R2 above 0.83) are high-fitting with the experimental results.  
There, the modulus elasticity was raised by increasing the SF content as well as that of NS and SP. This could be attributed to the improvement of compressive strength due to the  
Figure 14: Effect of fibre on the initial modulus of elasticity of HSC containing NS and SP (**a**) SF and (**b**) PPF.  
Figure 13: Compressive–tensile strength relationship of PPF reinforced HSC containing NS and SP.  
Figure 15: Effect of fibre on the secant modulus of elasticity of HSC containing NS and SP (**a**) SF and (**b**) PPF.  
bridging role of the fibres. [PERSON] et al. [65] performed SEM of the interfacial transition zone (ITZ) and showed a strong crystal bond between concrete paste and aggregate when nanoparticles were incorporated. Besides, it showed that the modulus of elasticity of concrete improved by 6% and 11% when 1% and 1.5% NS was added. It also showed that the use of a large NS content adversely affected the modulus of elasticity of concrete. Also, the improvement effect of NS on the initial elasticity modulus raised by increasing the amount of SP There, this property improved by 15%, 20%, 36%, 42% and 50% when only 0.1%, 0.2%, 0.3%, 0.4% and 0.5% SF were added, respectively. Furthermore, using a combination of 2% NS and 10% SP with 0.1%, 0.2%, 0.3%, 0.4% and 0.5% SF led to enhancing this property by approximately 55%, 62%, 82%, 92% and 102%, respectively. On the other hand, a similar trend can be observed for the secant modulus of elasticity; naturally, the value of the secant modulus of elasticity is lower than that of the initial modulus of elasticity in each concrete mix.  
Using PPF improves the value of the initial modulus of elasticity. Almost the same improvement in the modulus of elasticity due to PPF incorporation was reported by previous studies [61]. [PERSON] et al. [66] measured the effect of PPF on the mechanical characteristics of conventional concrete. They showed that the use of 0.15% and 0.3% PPF improved the modulus of elasticity of plain concrete respectively by 15% and 20%, in comparison with the control sample. They also argued that PPF bridges the cracks and transfer stress over cracking regions. In addition, PPF creates a strong concrete core at the middles of cylindrical specimens, which led to increasing the stress resistance and reducing the lateral and axial strain of samples that led to improving the modulus of elasticity of fibres-reinforced HSC [61]. Moreover, employing NS and SP resulted in improving the modulus of elasticity. As seen in these figures, this property raised around 7%, 10%, 15%, 21% and 27% by using 0.1%, 0.2%, 0.3%, 0.4% and 0.5% PPF, respectively. Besides, incorporating 2% NS with 10% SP and 0.1%, 0.2%, 0.3%, 0.4% and 0.5% PPF led to enhancing this property by 44%, 48%, 55%, 63% and 71%, respectively. According to Table 8, the influence of SF on increasing the content of the moduli of elasticity of HSC is greater than that of PPF.  
A high accuracy relationship between the moduli of elasticity and compressive strength was obtained, as illustrated in Figures 16 and 17. The proposed equations were established according to the type of fibres with R2 above 0.79 for the initial modulus of elasticity and 0.45 for the secant elasticity modulus. As seen in Figure 16, the slope of the relationship between the initial modulus of elasticity and compressive strength of PPF reinforced HSC was higher than that of SF reinforced HSC; however, an inverse trend is found for the secant modulus of elasticity (Figure 17).  
Figure 16: Effect of the fibres on the initial modulus of elasticity of HSC (**a**) SF and (**b**) PPF.