# Influence of Fatigue Crack Formation and Propagation on Reliability of Steel Members
## 3 Crack Propagation from Surface
stress-intensity factor \(\mathrm{K_{I}}\) for semi-elliptical surface crack propagating from the surface in a slab member subjected to tension and bending has the basic form [34; 35]  
\[\mathrm{K_{I}}=\mathrm{F(a)}\cdot\sqrt{\frac{\pi\cdot a}{\Phi}}\cdot(\sigma_{m}+ \mathrm{H}\cdot\sigma_{\mathrm{b}}), \tag{5}\]  
where:  
\(\mathrm{F(a)}=\mathrm{F(a/t,a/c,2c/b)}\) is the boundary-correction factor for surface crack in a plate,  
\(\mathrm{a,b,c,t,\theta}\) are the geometrical parameters of a member and a crack (Figures 1 and 2),  
\(\mathrm{a}\) is the crack depth,  
\(\mathrm{c}\) is the half of the crack width,  
\(\mathrm{b}\) is the width of the cracked plate,  
\(\mathrm{t}\) is the thickness of the cracked plate,  
\(\mathrm{\theta}\) is the parametric angle of ellipse,  
\(\mathrm{H}\) is the function expressing change of \(\mathrm{F(a)}\) in the case of bending stressing,  
\(\mathrm{\Phi}\) is the shape factor for an elliptic crack, where  
\[\mathrm{\Phi}=\int\limits_{0}^{\pi/2}\left[\left(\frac{a}{c}\right)^{2}\cdot \sin^{2}\theta+\cos^{2}\theta\right]^{0.5}\mathrm{d}\theta. \tag{6}\]  
One of the most used empirical Equation for \(\mathrm{\Phi}\) was derived in [36] and it depends on ratio between the crack depth \"a\" and half of crack width \"c\"  
\[\mathrm{\Phi}=1+1.464\cdot\left(\frac{a}{c}\right)^{1.65},\text{ for }\frac{a}{c}\leq 1, \tag{7}\]  
\[\mathrm{\Phi}=1+1.464\cdot\left(\frac{c}{a}\right)^{1.65},\text{ for }\frac{a}{c}>1. \tag{8}\]  
In general, it is possible to write the boundary-correction factor \(\mathrm{F(a)}\) in the following form  
\[\mathrm{F(a)}=\mathrm{M}\cdot\mathrm{S}=\left[\mathrm{M_{1}}+\mathrm{M_{2}} \cdot\left(\frac{a}{t}\right)^{2}+\mathrm{M_{3}}\cdot\left(\frac{a}{t}\right) ^{4}\right]\cdot\mathrm{S}, \tag{9}\]  
where \(\mathrm{M_{i}}\) are the curve-fitting functions,  
\[\mathrm{A_{tr}(N)}=\frac{1}{2}\cdot\pi\cdot a(N)\cdot c(N)\ [m^{2}]. \tag{2}\]0  
\[\mathrm{A_{tr}(N)}=\frac{1}{2}\cdot\pi\cdot a(N)\cdot c(N)\ [m^{2}]. \tag{2}\]1  
\[\mathrm{A_{tr}(N)}=\frac{1}{2}\cdot\pi\cdot a(N)\cdot c(N)\ [m^{2}]. \tag{2}\]2  
\[\mathrm{A_{tr}(N)}=\frac{1}{2}\cdot\pi\cdot a(N)\cdot c(N)\ [m^{2}]. \tag{2}\]3  
where:  
\(\mathrm{g}\) is the fine-tuning curve-fitting function,  
\(\mathrm{f_{\phi}}\) is the angular function derived from embedded elliptical crack solution,  
\(\mathrm{f_{w}}\) is the finite-width correction,  
and it is valid for \(\frac{a}{c}\leq 1\)  
\[\mathrm{A_{tr}(N)}=\frac{1}{2}\cdot\pi\cdot a(N)\cdot c(N)\ [m^{2}]. \tag{2}\]4  
\[\mathrm{A_{tr}(N)}=\frac{1}{2}\cdot\pi\cdot a(N)\cdot c(N)\ [m^{2}]. \tag{2}\]5\[\mathrm{A_{tr}(N)}=\frac{1}{2}\cdot\pi\cdot a(N)\cdot c(N)\ [m^{2}]. \tag{2}\]6  
If the focus is on biaxially symmetric welded steel I-beams and one assumes that there is a crack in the tension bottom flange (positive moment), then we can neglect the gradient of bending stress through flange thickness. In that case, it could be considered that the flange is subjected only to axial tensile force, from which it is stressed only by tensile (membrane) stress \(\sigma_{\mathrm{m}}=\sigma\). Then, it is possible to simplify Equation (5) into form  
\[\mathrm{A_{tr}(N)}=\frac{1}{2}\cdot\pi\cdot a(N)\cdot c(N)\ [m^{2}]. \tag{2}\]7  
However, this assumption should be only applied to thin flanges. In the case of thick flanges according to literature [35], the influence of the normal stress gradient on the thickness of the flanges is positively affected. In the case of longitudinal girders, the thickness of flanges varies from10 mm to 25 mm as maximum, whereas minimum height of longitudinal girder is 500 mm. The thickness of cross-girder flanges varies from 20 mm to 30 mm, whereas the cross-girders heights are minimum of 700 mm.