# The Application of Structural Reliability and Sensitivity Analysis in Engineering Practice
## 3 Case Study  
The possibilities of calculating various sensitivity measures in reliability analysis will be presented using the example of a cantilever beam. For this purpose, a beam with a length l = 150 cm, a square tube-shaped section with dimensions D = 8.0 cm and d = 6.0 cm, Young's modulus E = 21,000 kN/cm\({}^{2}\), Poisson's ratio v = 0.3, and yield strength fy = 23.5 kN/cm\({}^{2}\) is used. The beam load was assumed to be uniformly distributed with an intensity of q = 0.06 kN/cm, as shown in Figure 5. The values are the mean values of the random variables D, d, E, q, l, fy (Table 1). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Random Variable** & **Mean Value** & **Standard Deviation** & **Coefficient of Variation** \\ \hline D & 8 cm & 0.16 cm & 2\% \\ \hline d & 6 cm & 0.12 cm & 2\% \\ \hline E & 21,000 kN/cm\({}^{2}\) & 630 kN/cm\({}^{2}\) & 3\% \\ \hline q & 0.06 kN/cm & 0.0012 kN/cm & 2\% \\ \hline l & 150 cm & 3 cm & 2\% \\ \hline fy & 23.5 kN/cm\({}^{2}\) & 1.88 kN/cm\({}^{2}\) & 8\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Description of random variables. Figure 5: Geometry and load of the cantilever beam. The next step was to calculate the reliability index, failure probability, Bayesian credible interval of failure probability, and first-order Sobol indices using the Monte Carlo method. Below is a description of the results obtained in COMREL. Monte Carlo Sampling \(\mid\) Statistics  
Total number of samples: 1,000,000  
Total number of failed samples: 52,502  
Beta reliability index: 1.621  
Estimated probability of failure: \(5.3\times 10^{-2}\)  
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Bayesian credible intervals  
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upper bound  
probability that the actual probability of failure does not exceed the indicated value  
50%: p_f 5.3 \(\times\) 10\({}^{-2}\)  
75%: p_f 5.3 \(\times\) 10\({}^{-2}\)  
90%: p_f 5.3 \(\times\) 10\({}^{-2}\)  
95%: p_f 5.3 \(\times\) 10\({}^{-2}\)  
equal tail  
probability that the actual probability of failure is within the specified interval  
50%: [\(5.2\times 10^{-2}\) p_f 5.3 \(\times\) 10\({}^{-2}\)]  
75%: [\(5.2\times 10^{-2}\) p_f 5.3 \(\times\) 10\({}^{-2}\)]  
90%: [\(5.2\times 10^{-2}\) p_f 5.3 \(\times\) 10\({}^{-2}\)]  
99%: [\(5.2\times 10^{-2}\) p_f 5.3 \(\times\) 10\({}^{-2}\)]  
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Variance based sensitivity analysis  
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first-order sensitivity index  
Mean of LSF values: \(2.1\times 10^{-1}\)  
Figure 8: Graphical illustration of the elasticity of the reliability index based on standard deviation for SLS. \begin{table}
\begin{tabular}{l c} \hline \hline  & **FORM** & **SORM** \\ \hline reliability index & 4.434 & 4.388 \\ \hline failure probability & \(4.62\times 10^{-6}\) & \(5.73\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Estimation of reliability index and failure probability for ultimate limit state function. Figure 9: Graphical illustration of the coordinates of vector \(\alpha\) for ULS. The next step was to calculate the reliability index, failure probability, Bayesian credible interval of failure probability and first-order Sobol indices using the Monte Carlo method. Below is a description of the results obtained in COMREL. Monte Carlo Sampling! Statistics  
Total number of samples: 1,000,000  
Total number of failed samples: 8  
Beta reliability index: 4.388  
Estimated probability of failure: 8.0 \(\times\) 10\({}^{-6}\)  
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Bayesian credible intervals  
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upper bound  
probability that the actual probability of failure does not exceed the indicated value 50%:  
P_f 8.7 \(\times\) 10\({}^{-6}\) 75%:  
P_f 1.1 \(\times\) 10\({}^{-5}\) 90%:  
P_f 1.3 \(\times\) 10\({}^{-5}\) 95%:  
P_f 1.4 \(\times\) 10\({}^{-5}\) 99%:  
P_f 1.7 \(\times\) 10\({}^{-5}\)  
Figure 11: Graphical illustration of the elasticity of the reliability index based on standard deviation for ULS.