# Influence of Fatigue Crack Formation and Propagation on Reliability of Steel Members
## 6 Independent Crack Growth in Directions \"a\" and \"c\"
### Probabilistic Approach  
The probabilistic approach was used for finding the impact of fatigue crack on failure probability variation and on corresponding reliability index [42]. In that case, all strength and geometrical characteristics were considered as random variables with normal distribution N(m,s\({}^{2}\)) and they are shown in Table 2[43; 44]. The steel of quality S235 was assumed. Statistical characteristics of yield strength were taken from [45]. Likewise, all geometric characteristics (thickness and width) of plates were taken from [45]. Standard deviations, reflecting tolerances of flange widths, take into account the deviations allowed by standard [46]. Dimensions of crack semi-axes (c,a) were again chosen and were considered as constants, not as random variables. Numerical application of resistance calculation R = M\({}_{\text{pl,y}}\) was realised using the simulation method Monte-Carlo. Figure 5: Influence of crack size on plastic section modulus W\({}_{\text{pl,y}}\). Obtained results of mean value \(\rm m_{R}\) and standard deviation \(\rm s_{R}\) variation are shown in Figures 7 and 8. As expected, the results show detectable decrease of the mean value \(\rm m_{R}\) and standard deviation \(\rm s_{R}\). This is the proof that influence of fatigue crack development and propagation may have unavoidable influence on member resistance. In this case, the mean value of resistance \(\rm m_{R}\) was decreased by a maximum 15.33% (I 400) and 19.54% (I 800) and the standard deviation \(\rm s_{R}\) was decreased by a maximum 12.73% (I 400) and 18.73% (I 800). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Input Characteristics**} & \multicolumn{3}{c}{**I 400-Welded**} & \multicolumn{3}{c}{**I 800-Welded**} & \multicolumn{3}{c}{**Probability**} \\ \cline{2-7}  & **m** & **s** & **v** & **m** & **s** & **v** & **Distribution** \\ \hline Yield strength—\(\rm f_{y}\) [MPa] & 269.20 & 28.00 & 0.104 & 281.00 & 27.80 & 0.099 & Normal \\ \hline Flange width—\(\rm b_{f}\) [mm] & 100.00 & 2.00 & 0.020 & 250.00 & 2.00 & 0.008 & Normal \\ \hline Flange thickness—\(\rm t_{f}\) [mm] & 10.210 & 0.37 & 0.036 & 25.525 & 0.925 & 0.036 & Normal \\ \hline Web height—d [mm] & 380.00 & 2.00 & 0.005 & 750.00 & 2.00 & 0.003 & Normal \\ \hline Web thickness—\(\rm t_{w}\) [mm] & 8.168 & 0.296 & 0.036 & 10.210 & 0.37 & 0.036 & Normal \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistical characteristics of welded steel beams I 400 and I 800. Figure 6: Influence of crack size on plastic resistance moment \(\rm M_{pl,y}\). The consequential obtained values of parameters are  
\[\text{I 400: m}_{\text{R}}=186.515\text{ kNm},\text{ I 800: m}_{\text{R}}=1792.906\text{ kNm},\]  
\[\text{s}_{\text{R}}=20.176\text{ kNm},\text{ s}_{\text{R}}=186.036\text{ kNm}\]  
The load effects E were considered as random variables normally distributed with the following parameters  
\[\text{I 400: m}_{\text{E}}=94.411\text{ kNm},\text{ I 800: m}_{\text{E}}=943.651\text{ kNm},\]  
\[\text{s}_{\text{E}}=15.132\text{ kNm},\text{ s}_{\text{E}}=139.527\text{ kNm}.\]  
From parameters follows that members were designed with reliability index \(\beta=3.652\) for design lifetime \(\text{T}_{\text{d}}=100\) years. The results of crack growth influence on failure probability \(\text{P}_{\text{f}}\) and reliability index \(\beta\) are shown in Figures 9 and 10. Decrease of the mean value \(\text{m}_{\text{R}}\) causes decrease of the member's--decrease of reliability index \(\beta\) and corresponding increase of failure probability \(\text{P}_{\text{f}}\). Decrease of the standard deviation, in general, has favourable influence on members' reliability (the failure probability \(\text{P}_{\text{f}}\) is lower in time). Since the results show a decrease of the reliability, it follows that the reliability is mainly influenced by the decrease of the mean value \(\text{m}_{\text{R}}\). The decrease of the  
Figure 7: Influence of crack size on resistance mean value \(\text{m}_{\text{R}}\).