# Safety Concept for Textile-Reinforced Concrete Structures with Bending Load
## 5 Reliability Calculation Results
### Influence of the Effective Depth \(d\)  
The influence of the effective depth \(d\) on the reliability index \(\beta\) was investigated for a carbon textile reinforcement with effective depths of \(d\) = 15 mm and \(d\) = 60 mm (Figures 12 and 13). The effective depth \(d\) has an influence on the progression of the reliability index curves both in state A and in state C. Subsequently, the effective depth \(d\) is 60 mm; to achieve an appropriate safety level, it is sufficient to apply a partial safety factor \(\gamma_{\rm T}\) equal to 1.3 and a partial safety factor \(\gamma_{\rm C}\) equal to 1.5 in states A and C, respectively. For the smallest effective depth, i.e., \(d\) = 15 mm, and for the selected partial safety factors \(\gamma_{\rm T}\) equal to 1.3 and \(\gamma_{\rm C}\) equal to 1.5, the required reliability index \(\beta_{\rm target}\) of 3.8 cannot be ensured. These results can be explained by the large influence of the effective depth \(d\) scattering on the reliability index \(\beta\), which is proven by the sensitivity factors (Figure 13). These observations are valid for the application of a linear or a rectangular stress-strain curve for the concrete analysis [9]. Figure 11: Sensitivity factors calculated for \(\gamma_{\rm T}\) = 1.3 according to FORM: (a) AR-glass textiles; (b) carbon textiles. Figure 12: Reliability indices calculated according to FORM: (a) effective depth \(d\) = 15 mm; (b) effective depth \(d\) = 60 mm. In the case of very thin structural members, even a slight deviation of the reinforcement from the correct position can lead to a large reduction of the bending capacity. For this reason, in order to ensure a sufficient safety level with a constant partial safety factor \(\gamma_{\text{T}}\) independent of the effective depth, the authors recommend using a reduction factor for small effective depth, taken from the old German code DIN 1045(88) [30]. DIN 1045(88) proposes to increase the internal forces with a factor of 15/(\(d+8\)) for structural members with effective depths smaller than 70 mm. This approach can be adopted for thin textile-reinforced members. Since the probabilistic calculations show that the required safety level can be ensured for effective depths \(d\) greater than 60 mm (Figure 12a), it is sufficient to apply a reduction for effective depths \(d\leq 60\) mm. With Equations (17) and (18), an effective depth \(d_{\text{eff}}\) can be determined, which can be used to calculate the required flexural textile reinforcement. \[\text{for }d\leq 60\text{ mm }d_{eff}=\frac{d-4}{0.95}\text{ (d in mm)} \tag{17}\]  
\[\text{for }d>60\text{ mm }d_{eff}=d \tag{18}\]  
Figure 14 shows the influence of \(d\) and \(d_{\text{eff}}\) on the reliability index. The required safety level is ensured with the effective depth \(d_{\text{eff}}\) for the three states A, B and C. In contrast, in the calculation of the reliability indices, the threshold \(\beta_{\text{target}}\) of 3.8 cannot be achieved when the effective depth d is not reduced. Consequently, the required safety level can be ensured while retaining the common partial safety factor for concrete \(\gamma_{\text{C}}\) equal to 1.5. Figure 13: Sensitivity factors calculated for \(\gamma_{\text{T}}=1.3\) according to FORM: (**a**) effective depth \(d=15\) mm; (**b**) effective depth \(d=60\) mm. Summary and Proposal for the Partial Safety Factor for Textile Reinforcement \(\gamma_{\mathrm{T}}\)  
The probabilistic calculations show that the safety level depends significantly on the reinforcement ratio and on the effective depth. These two parameters primarily influence the required partial safety factors for textile reinforcement \(\gamma_{\mathrm{T}}\) and for concrete \(\gamma_{\mathrm{C}}\). The remaining parameters--tensile strength of the textile reinforcement, concrete compressive strength, long-term durability factor \(\alpha_{\mathrm{t}}\) and the ratio of dead to live loads--seem to have insignificant influence on the probability of failure and, therefore, on the required partial safety factors.