## 4 Theoretical Approach and Discussion  
### Material Behavior  
The _fib_ Model Code 2010 [34] presents the most comprehensive design guidelines for fiber-reinforced concrete, focusing both on material behavior and structural behavior. The basic principles introduced are mainly obtained from research of steel fiber reinforced concrete (SFRC), but the _fib_ model is adaptable to other types of fibers [37]. SFRC exhibits strain-softening behavior, as shown in Figure 6. Simplified load-deformation curves; (a) strain softening, and (b) strain hardening [37]. However, this strain softening-behavior can be different for FRC incorporating synthetic fibers.  
Figure 5: (a) Fibers bridging the crack; (b) Zoom on the wider section of the crack.  
Figure 6: Simplified load-deformation curves; (a) strain softening, and (b) strain hardening [37].  
A typical load vs. CMOD curve for plain concrete and SFRC is depicted in Figure 7a. While comparing against Figure 4 it is evident that even though both SFRC and twisted-bundle macro-synthetic fiber FRC demonstrate strain-softening behavior, the decrease in load value after CMOD1 = 0.5 mm is slightly different. For instance, for SRFC F1 \(>\) F2 \(>\) F3 \(>\) F4, as seen in Figure 7a, with the typical stress distribution shown in Figure 7b. However, for the twisted-bundle macro-synthetic fiber-reinforced SCC, the load remains almost constant up to CMOD3 = 2.5 mm, followed by a softening behavior.  
In order to predict the moment capacity of a FRC element under bending, the material behavior, particularly the stress-strain behavior of FRC, should be defined first. The loads in Figure 4 can be converted to flexural and residual flexural strengths, using Equations (1) and (2). Figure 8 displays the stress vs. CMOD curves (dashed lines) along with simplified curves connecting maximum stress, stress corresponding to CMOD1, CMOD2, CMOD3, and CMOD4 (solid lines). It can be noted here that the CMOD1 corresponds to the serviceability limit state, whereas the CMOD3 usually corresponds to the ultimate limit state [37]. Since the stress remained almost constant between CMOD1 to CMOD3, it can be assumed that \(f_{R1}=f_{R3}\), and a simplified model can be proposed, as shown in Figure 9. In addition, it can be seen that the drop in magnitude from \(f_{L}\) to \(f_{R1}\) is less as the fiber dosage increases from 4 kg/m\({}^{3}\) to 8 kg/m\({}^{3}\). The \(f_{R1}/f_{L}\) ratio of the four samples of the three different mixtures are tabulated in Table 8. \(f_{R1}/f_{L}\) ratio of all tested samples for three different mixtures, the average and standard deviation of each mixture are also presented. The ratio is within an acceptable standard deviation for FF4 and FF8, while FF6 exhibited a significant variation. Even though the variation for FF6 is high, it can be assumed as a general trend that the \(f_{R1}/f_{L}\) ratio increases with the increase in fiber dosage.  
Figure 7: (**a**) Typical load vs. CMOD curve for plain concrete (in grey) and fiber-reinforced concrete (FRC) (in black) and (**b**) Stress distributions assumed for FRC in the _fib_ Model [37].  
Figure 8: Flexural stress vs. CMOD plots for (**a**) FF4, (**b**) FF6 and (**c**) FF8, along with the corresponding line connecting maximum stress to stress at CMOD1, CMOD2, CMOD3, and CMOD4.