## 5 Case Study: SFRC Beam
### Sensitivity Analysis  
Figures 14-20 present the total Sobol' indices for the forces and displacements of three characteristic points of the considered SFRC beams' flexural behavior. Higher total Sobol' index of a material property indicates that the variance of that material property has a higher effect on the variance of the considered QoI, and therefore, changing the value of that material property will have a larger effect on the considered QoI. Material properties with low total Sobol' indices, can be set to a constant value (e.g., to their expected values) without significantly changing the variance of the considered QoI; therefore, the change in the values of material properties with low Sobol' indices will not have a large effect on the considered QoI. This property can be used for dimensionality reduction [42]. Figures 14 and 15 present total Sobol' indices for first-crack force and displacement. The tensile strength of the SFRC has the highest total Sobol' index of around 0.8. The variance of the residual tensile strength contributes around 20% to the variance of first-crack force and around 10% to the variance of the first-crack displacement. Furthermore, the variance of the elastic modulus of the SFRC is responsible for around 20% of variance of the first-crack displacement but has a very small effect on the first-crack force. Figure 14: Total Sobol’ indices for first-crack force. Figure 13: Histogram of 500 samples of ductility factor values obtained using the numerical model and PCE metamodel. Figures 16 and 17 present total Sobol' indices for yield force and yield displacement. The variance of yield force is dominated by the variance of the SFRC's residual tensile strength, accounting for around 90% of the variance. The variance of the yield strength of the bar reinforcement contributed only 10% to the variance of the yield force. However, despite the relatively low CoV of rebar yield strength of 5%, its variance contributed to 45% of the variance of yield displacement, indicating that rebar yield strength is very important for the yield displacement of the SFRC beam. The variance of residual tensile strength, SFRC's elastic modulus, and tensile strength were also recognized as important sources of uncertainty in the case of the yield displacement. Figure 16: Total Sobol’ indices for yield force. Figure 15: Total Sobol’ indices for first-crack displacement. Total Sobol' indices for collapse force and collapse displacement are presented in Figures 18 and 19. Similarly as in the case of yield force, the variance of the collapse force was primarily caused by the variance of the SFRC's residual tensile strength. The variance of the collapse displacement is also dominated by the variance of the SFRC's residual tensile strength. The variance of the SFRC's compressive strength was also important for the variance of the collapse displacement, probably due to certain cases in which beams achieved compression failure, as defined in Section 3.3. Figure 17: Total Sobol’ indices for yield displacement. Figure 18: Total Sobol’ indices for collapse displacement. Figure 20 presents total Sobol' indices for the ductility of the considered SFRC beam. As ductility is defined as the ratio between the collapse and yield displacement, the variance of the ductility depends on the variance of these two displacements. The variance of rebar yield strength and SFRC's residual tensile strength contribute the most to the uncertainty in the SFRC beam's ductility. This means that changing these two material properties is the most effective way to modify the beam's ductility. Thus, the most effective way of increasing the ductility of the SFRC beam considered in this case study is by increasing the residual tensile strength of SFRC (e.g., by increasing the fiber dosage), thereby increasing the collapse displacement by delaying the attainment of 2% strain in tension.