# Influence of Fatigue Crack Formation and Propagation on Reliability of Steel Members
## 5 Effect of Crack on Steel Member's Reliability  
In the engineering probability method, the reliability margin g(R,E) = G is given by Equation  
\[\mathrm{g(R,E)=G=R-E,} \tag{20}\]  
where:  
R is the random variable resistance of a structural member,  
E is the random variable load effects of the same element. It is assumed that the member resistance R is normally distributed random variable \(\mathrm{N(m_{R},s_{R}{}^{2})}\) and that the load effects E are also normally distributed \(\mathrm{N(m_{E},s_{E}{}^{2})}\), where is valid that R and E are mutually independent. The normally distributed random variables R and E are given by the mean values (m\({}_{R}\), m\({}_{E}\)) and the standard deviations (s\({}_{R}{}^{2}\), s\({}_{E}{}^{2}\)). The member's reliability is given either by the failure probability Pf or by reliability index \(\beta\). If one considers the change of member resistance R(N) in time t due to fatigue damage (crack formation and development depending on cycles number N), then the reliability of the member is also changing in time, it follows that the failure probability \(\mathrm{P_{f}(N)}\) and reliability index \(\beta(N)\) are also varying in time t.  
As was mentioned, the paper is focused on the welded steel biaxially symmetric I-beam subjected to bending. In this case, the member's resistance R(t) for compact and semi compact cross-sections is given by the equation  
\[\mathrm{R(N)=M(N)_{pl,y}=f_{y}\cdot W(N)_{pl,y},\ [kNm]\ for\ cross\-sections\ of\ class\ 1\ and\ 2,} \tag{21}\]  
\[\mathrm{R(N)=M(N)_{el,y}=f_{y}\cdot W(N)_{el,yd,},\ [kNm]\ for\ cross\-sections\ of\ class\ 3,} \tag{22}\]  
where:  
fy is the random variable yield strength of structural steel [MPa],  
\(\mathrm{W(N_{pl,y})}\) is the time dependent random variable plastic section modulus [m\({}^{3}\)],  
\(\mathrm{W(N_{el,y})}\) is the time dependent random variable elastic section modulus [m\({}^{3}\)]. That theoretical approach does not consider the cross-sections of class 4. In the case of compact cross-section 1 and 2, the time dependent plastic section modulus \(\mathrm{W(N)_{pl,y}}\) taking into account the cross-section loss due to crack development is represented by the relation  
\[ \begin{array}{l}\mathrm{W(N)_{pl,y}=b_{f}\cdot t_{f}\left(d-e_{I}^{*}(N)+ \frac{t_{W}}{2}\right)+\frac{t_{W}}{2}\cdot\left[\left(d-e_{I}^{*}(N)\right)^ {2}+\left(e_{I}^{*}(N)\right)^{2}\right]+}\\ +\left[b_{f}\cdot t_{f}-0.5\cdot\pi\cdot a(N)\cdot c(N)\right]\cdot\left[e_{I}^ {*}(N)+\frac{b_{f}\cdot t_{f}^{2}-\pi\cdot a(N)\cdot c(N)\cdot\left(b_{f}- \frac{4\cdot a(N)}{3\cdot\pi}\right)}{2\cdot b_{f}\cdot t_{f}-\pi\cdot a(N) \cdot c(N)}\right]\end{array}  \tag{23}\]  
where  
\[\mathrm{e_{I}^{*}(N)=\frac{d}{2}+\frac{\pi\cdot a(N)\cdot c(N)}{4\cdot t_{W}}, \ (\text{shows\ the\ change\ of\ cross-sectional\ centre\ of\ gravity}).} \tag{24}\]  
Meaning of each cross-sectional characteristics is shown in Figure 3.  
Similarly, the time dependent elastic section modulus \(W(N)_{el,y}\) of the cross-sections of class 3 taking into account the cross-section loss of the flange due to crack development is given by Equation  
\[W(N)_{el,y,hor}=\frac{I(N)_{y}}{d+t_{f}-e_{I}^{*}(N)},\text{ for the upper fibres of cross-section}, \tag{25}\]  
\[W(N)_{el,y,dol}=\frac{I(N)_{y}}{t_{f}+e_{I}^{*}(N)},\text{ for the bottom fibres of cross-section}, \tag{26}\]  
where the second moment of area is calculated according to  
\[ \begin{split}& I(N)_{y}=\frac{1}{6}\cdot b_{f}\cdot t_{f}^{3}+b_{f} \cdot t_{f}\cdot\left[\left(\frac{t_{f}}{2}+d-e_{I}^{*}(N)\right)^{2}+\left( \frac{t_{f}}{2}+e_{I}^{*}(N)\right)^{2}\right]+\frac{1}{12}\cdot t_{w}\cdot d ^{3}+\\ &+t_{w}\cdot d\cdot\left(e_{I}^{*}(N)-\frac{d}{2}\right)^{2}- \frac{\pi}{8}\cdot a^{3}(N)\cdot c(N)-\frac{\pi}{2}\cdot a(N)\cdot c(N)\cdot \left(e_{I}^{*}(N)+t_{f}-\frac{4\cdot a(N)}{3\cdot\pi}\right)^{2},\end{split}