# Probabilistic Studies on the Shear Strength of Slender Steel Fiber Reinforced Concrete Structures
## 6 Derivation of Partial Safety Factor for Resistance Model Uncertainties  
The Eurocode EN 1992-2 [36] and Fib Model Code 2010 [20] recommended various global partial factors \(\gamma_{Rd}\) for model uncertainty for nonlinear analysis. In the case of Eurocode, \(\gamma_{Rd}=1.06\), whereas for Fib Model Code 2010 \(\gamma_{Rd}=1.0-1.1\) (for models with zero to high uncertainties). However, various studies [35; 37] have shown that these values are inadequate and should be related to the specific failure mode under consideration.  
The partial safety factors for model uncertainty related to the MC-10, Bernat et al., and DAfStB are proposed in this section. The global safety assessment of an SFRC structure for the ultimate limit state is performed based on the design requirement that the design action \(E_{d}\) is lower than the design structural resistance \(R_{d}\) (See Equation (9)) [20; 38].  
\[E_{d}<R_{d} \tag{9}\]  
In line with the EN 1990 [38], the partial safety factor \(\gamma_{Rd}\) for model uncertainties is generally incorporated into the design equation through the expression in Equation (10) [38].  
\[R_{d}\ =R\bigg{[}\eta\frac{X_{k}}{\gamma_{m}};a_{d}\ldots\bigg{]}/\gamma_{Rd} \tag{10}\]  
where \(R_{d}\) is the design resistance. \(X_{k}\) is the characteristic values of the variable. \(\eta\) is a conversion factor appropriate to the material property. \(\gamma_{m}\) is the material property factor. \(a_{d}\) is the design geometric parameter. \(\gamma_{Rd}\) is global partial factor for resistance.  
Based on a lognormal probability distribution, the global partial safety factor representing the resistance model uncertainties \(\gamma_{Rd}\) can be obtained from Equation (11) [37]. For the shear predictive models investigated in this study, the probability distribution for the model uncertainty random variables is taken as a lognormal distribution based on the assessment presented in Section 5.4.  
\[\gamma_{Rd}=1/[\mu_{U}.exp.(-\alpha_{R}.\beta.V_{U})] \tag{11}\]  
where \(\alpha_{R}\) denotes the FORM sensitivity factor = \(0.8\times 0.4=0.32\)[38] (0.4 is the factor for non-dominant resistance variable and 0.8 is the sensitivity factor for resistance). \(\beta\) is the target reliability, according to EN 1990. \(\mu_{U}\) and \(V_{U}\) are the mean and coefficient of variation of the model uncertainty random variable, respectively.  
Figures 7 and 8 present the variation of the obtained model uncertainty global partial factor \(\gamma_{Rd}\) with target reliability \(\beta\) for the various models and the considered parametric range of shear span to effective depth ratio. The range of target reliability presented in the figures covers the target reliabilities for 50 years of design working life period for the ultimate limit states for structures. As shown in the figures, the model uncertainty factor \(\gamma_{Rd}\) increases as the target reliability \(\beta\) increases. The recommended model uncertainty partial factor \(\gamma_{Rd}\) obtained for the investigated models is presented in Table 6. The recommended value is obtained as the average of the partial safety factors for the range of shear span to effective depth ratio considered.  
Figure 8: Variation of the partial factor \(\gamma_{Rd}\) with \(\beta\) (using the statistics of the different subsets of shear span to effective depth ratio). (**a**) Short shear span length. (**b**) Medium shear span length. (**c**) Slender shear span length.  
Figure 7: Variation of the partial factor \(\gamma_{Rd}\) with \(\beta\) (using the statistics of the full database).