## 4 Considerations for Bending Design  
An interesting example where the concepts of plastic regions and admissible failure mechanisms can be applied to FRC is bending design. The topic of bending design has been investigated from various perspectives in the past. For instance, several approaches propose to consider an elastic state of stresses in the compression zone while a crack opens in bending [40]. Such approaches typically require considering that the region in compression (where elastic strains are considered) is associated with a fictitious length, linearly increasing with the opening of the flexural crack [40]. Although the global response can be satisfactorily reproduced when compared to test results, such consideration presents theoretical shortcomings. For instance, the value of the peak stress is not controlled, the local strains in the compression region remain relatively low with respect to what might be expected in reality, and the meaning of the fictitious length is not connected to a physical parameter. Figure 4: Load-deflection and stress-crack opening curves for the following: (**a**) Steel FRC with hooked-end fibers; (**b**) Macro-polymer FRC; (**c**) Hybrid FRC (polymer and hooked-end steel fibers); (**d**) Steel FRC with straight fibers; (**e**) Hybrid FRC with two geometries of fibers. An alternative approach to the flexural resistance and its associated kinematics at failure can be treated within the framework of limit analysis by considering a failure mechanism for plane strain conditions, as shown in Figure 5a. Such a mechanism consists of a separation line in tension (increment of plastic strains in pure tension) as well as a plastic wedge in compression (Rankine region). Such an approach has proven to be applicable to reinforced concrete sections under bending and normal forces [41] and has been successfully applied as the ground model for validation of the EN1992-1-1:2023 [29] design expressions. For its application to FRC, the stress distribution shown in Figure 5b applies, where the corresponding flexural strength results:  
\[M_{R}=f_{fp}\cdot b\cdot h^{2}\bigg{(}1-\frac{x_{pl}}{h}\bigg{)}\bigg{(}1-\frac {x_{pl}}{2h}\bigg{)} \tag{4}\]  
where \(x_{pl}\) refers to the height of the compression zone, whose value is determined by the following equilibrium conditions:  
\[x_{pl}=h\cdot\frac{f_{ct}}{f_{cp}+f_{fp}} \tag{5}\]  
This term can be approximated in most cases considering \(x_{pl}\to 0\) and thus  
\[M_{R}\cong\frac{1}{2}f_{fp}\cdot b\cdot h^{2} \tag{6}\]  
Figure 5: Flexural resistance according to limit analysis under plane strain conditions: (**a**) Plastic regions and separation lines; (**b**) Resulting stresses; (**c**) Kinematics of the failure mechanism; (**d**) Detail of strains and stresses in the plastic wedge; (**e**) Yield surface (plane strain conditions) and normality condition of increment of strains; (**f**) Rigid and plastic bodies after plastic rotation. The conditions of the plastic mechanism are investigated in detail in Figure 5c-f. By assuming a rigid body motion at the tip of the separation yield line, the deformed shape of Figure 5c results. In the resulting plastic wedge (Figure 5d), the strain state can be obtained by pure kinematic conditions, where the increment of principal compressive strain results \(\dot{\epsilon_{2}}=-\dot{\psi}/4\) while the corresponding increment of the principal tensile strain results \(\dot{\epsilon_{1}}=\dot{\psi}\). As can be noted, the ratio between the increment of tensile and compressive strains results in 4, which corresponds to the one associated with the normality condition to the yield surface (\(4\cong(1+\sin\varphi)/(1-\sin\varphi)\) for plane strain conditions, see Figure 5e). Such a state of strains leads the plastic wedge to an axial shortening while there is a transversal expansion, typically associated with longitudinal cracking (Figure 5f). The validity of such considerations of the failure mechanism for FRC can be easily demonstrated with the help of available tests.