# Reliability Assessment of Steel-Lined and Prestressed FRC Slabs against Projectile Impact
## 5 Data for Reliability Analysis  
Prestressed and steel-lined fiber-reinforced concrete (PFRC) slabs are subject to substantial uncertainties, so we identified those variables that contain substantial uncertainties and considered them as random, as shown in Table 2. The bias factor in this table is calculated by dividing the mean value of the random variable by its nominal value (which is fixed in a deterministic sense). The nominal value and the mean value are the same when this factor is 1. Bias factors are greater than one for resistance-related variables. For variables related to loading, this factor is less than 1.0.  
Figure 3: Details of steel-lined RC slabs (All dimensions are in mm).  
Reliability analysis of PFRC slab specimens also requires probability distributions of the expected extreme impact load, measured in terms of impact velocity. In this study, projectile velocity was described by the Extreme Type I distribution (Table 2). This distribution's probability density function (PDF) and CDF are given by [PERSON] and [PERSON] [17]:  
\[\mathrm{PDF}:f(x)=\alpha\exp\Bigl{\{}-e^{-\alpha(x-u)}\Bigr{\}}\exp\{-\alpha(x- u)\} \tag{12}\]  
\[\mathrm{CDF}:F(x)=\exp\Bigl{\{}-e^{-\alpha(x-u)}\Bigr{\}}\text{ for }-\infty\leq x\leq\infty \tag{13}\]  
Here, \(\alpha\) and \(u\) represent distribution parameters. In order to estimate distribution parameters, [PERSON] and [PERSON] [17] proposed the following formulas:  
\[\alpha\approx\frac{1.282}{\sigma_{x}} \tag{14}\]  
\[u\approx\mu_{x}-0.45\sigma_{x} \tag{15}\]  
in this equation, \(\mu_{x}\) represents the mean and \(\sigma_{x}\) represents the standard deviation. References for the probability distributions and coefficients of variation are given in Table 2.  
It is critical to know the required number of simulations when performing a reliability analysis with MCS. An example of how \(COV\Bigl{(}P_{f}\Bigr{)}\) varies with the simulation number shown in Figure 4. This slab specimen (RE-SL-F2-P1) was impacted by the projectile at a nominal strike velocity of 185 m/s. Figure 4 shows that \(COV\Bigl{(}P_{f}\Bigr{)}\) decreases with increasing simulation cycles (N). For the present specimen, N greater than 1,000,000 (1 million) gives an adequate \(P_{f}\) (i.e., \(COV\Bigl{(}P_{f}\Bigr{)}\) smaller than 5%). In the current study, the MCS was performed with enough simulations for \(COV\Bigl{(}P_{f}\Bigr{)}\) to be less than 5%.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Random Variable** & **Nominal** & **Bias Factor** & \
\begin{tabular}{c} **COV** \\ **(Coefficient** \\ **of Variation)** \\ \end{tabular} &
\begin{tabular}{c} **Probability** \\ **Distribution** \\ \end{tabular} & **Reference** \\ \hline \multicolumn{6}{c}{**PFRC slab**} \\ \hline Density of concrete, \(\rho_{c}\) (kg/ m\({}^{3}\)) & 2500 & 1.05 & 0.10 & Lognormal & [19] \\ \hline Effective tensile strength of concrete, \(f_{teff}\) (MPa) & Variable & 1.10 & 0.10 & Lognormal & Assumed \\ \hline Thickness of slab, \(H\) (mm) & 100 & 1.00 & 0.05 & Normal & [19] \\ \hline Reinforcement ratio, \(r\) (\%) & Estimated \({}^{\$}\) & 1.10 & 0.10 & Normal & Assumed \\ \hline Steel rebar spacing, \(c_{r}\) (mm) & 100 & 0.90 & 0.05 & Lognormal & [16] \\ \hline Reinforcing index. \(R1\) & Estimated \({}^{\$}\) & 1.10 & 0.12 & Lognormal & [16] \\ \hline \multicolumn{6}{c}{**Hemispherical nose projectile**} \\ \hline Projectile mass, \(M\) (kg) & 0.80 & 1.10 & 0.05 & Lognormal & [20] \\ \hline Projectile diameter, \(d\) (mm) & 40.0 & 1.05 & 0.05 & Normal & [20] \\ \hline Projectile strike velocity, \(V_{0}\) (m/s) & Variable & 0.90 & 0.10 & Extreme Type I & [16] \\ \hline \end{tabular} \({}^{\$}\) Nominal value was estimated using the appropriate equation.  
\end{table}
Table 2: Random variables and statistical data.