# Influence of Fatigue Crack Formation and Propagation on Reliability of Steel Members
## 7 Dependence of Crack Growth in Directions \"a\" and \"c\"  
In fact, the crack growth in one direction depends on the crack growth in the other. Although this dependency was considered in several works such as [48; 49; 50], it has not yet been precisely defined. Approach described in [51] was used for obtaining this dependency. For simplification, the crack growth is analysed at points (directions) \"a\" and \"c\" (in direction of main semi-axes). Crack propagation speed at those points is described by the stress-intensity factor (\(\Delta\)K) and by the Paris-Erdogan law, which can be expressed as  
\[\frac{\mathrm{da}}{\mathrm{dN}}=\mathrm{C}\cdot(\Delta\mathrm{K_{a}})^{\mathrm{ m}}, \tag{30}\]  
\[\frac{\mathrm{dc}}{\mathrm{dN}}=\mathrm{C}\cdot(\Delta\mathrm{K_{c}})^{ \mathrm{m}},\ \mathrm{respectively}, \tag{31}\]  
where:  
\(\Delta\mathrm{K_{a}}\) is the stress-intensity factor in direction \"a\",  
\(\Delta\mathrm{K_{c}}\) is the stress-intensity factor in direction \"c\",  
C, m are the material constants. Figure 10: Influence of crack size on reliability index \(\beta\). increments \(\Delta\)c and \(\Delta\)a are added to previous dimensions. Again, choosing increment \(\Delta\)c (same as before) and calculating a new corresponding increase \(\Delta\)a, new crack dimensions can be calculated. Calculation of crack growth will be finished if none of the restrictions (29) is violated. An example of crack dimension calculation for flange thickness \(\mathrm{t_{f}}=\mathrm{t}=10\) mm is shown in Table 3 and the results are shown in Figure 11.  
The initial ratio a/c (initial crack dimensions) is important for crack growth calculation in both directions. Basically, it is the minimum detectable crack. Since the minimum value of the measurable crack is not precisely defined (usually it is estimated) and similarly the ratio a/c is not known, the approach was intended for values close to zero (exact zero cannot be considered because of the ratio a/c). The work [29] is also concerned with this problem. The results show that the increment \(\Delta\)a is gradually increasing unevenly at constant increment \(\Delta\)c.  
It is possible to obtain the crack growth in both directions using the described approach. The curve was drawn through the obtained values based on finding the dependence of crack growth in direction \"c\" on crack growth in direction \"a\". The second degree polynomial curve turned out to be the most preferred form  
\[\mathrm{c}=\mathrm{k_{i}}\cdot\mathrm{a^{2}}+\mathrm{k_{2}}\cdot\mathrm{a}+ \mathrm{k_{3}}  
which improved values of the coefficient \(\rm k_{i}\). Table 4 shows the \(\rm k_{i}\) values depending on the selected increment \(\Delta c\), wherein the thickness of the flange was assumed \(\rm t_{f}=t=10\) mm.  
It is assumed that the increment reduction is meaningless and the values of coefficients \(\rm k_{i}\) are not significantly changed. Therefore, the dependency for \(\Delta c=0.001\) mm is considered to be sufficiently accurate.  
Similarly, the parametric study was used for investigation of the coefficients \(\rm k_{i}\), which are valid to other flange thickness \(\rm t_{f}\). Dependencies are shown in Table 5.  
If the dependency is established of \"c\" not only on \"a\", but also on flange thickness \(\rm t=t_{f}\) in the approach, the universal dependency of the crack growth in direction \"c\" on direction \"a\" and on flange thickness \(\rm t\), is obtained in the form  
\[\rm c=\frac{0.3027}{t}\cdot a^{2}+1.0202\cdot a+0.00699\cdot t. \tag{42}\]  
Again, the parametric study with same steel welded I-beams I 400 and I 800, which were considered before, was performed. The characteristics are given in Table 2.  
The achieved results of change of the mean value \(\rm m_{R}\) and the standard deviation \(\rm s_{R}\) are shown in Figures 12 and 13.