# Retrofit Fatigue Cracked Diaphrag Cutouts Using Improved Geometry in Orthotropic Steel Decks
## 5 Results and Discussion
### Fatigue Life Assessment of Diaphragm Cutouts  
The cutout retrofit method of hot cutting combined with edge-corner grinding causes residual tensile stress, which affects the stress ratio of the wheel load induced stress spectrum. But the stress ratio has little effect on the fatigue life of the whole steel structure, since fatigue life is mainly related to stress amplitude. Therefore, the effect of residual cutting stress on fatigue life is ignored in this paper. In Eurocode 3 (\"Design of steel structures\" (Parts 1-9)), the fatigue analysis method named the Vierendeel-model is presented to calculate the fatigue life of diaphragm cutouts, which ignores the effect of the cutout geometry. Therefore, it is not suitable to evaluate the fatigue life of cutouts in this paper. Another approximate assessment method is mentioned in Eurocode 3, in which the modified nominal stress is used to evaluate fatigue life for a geometric discontinuity that has not been considered in the classification of a particular construction detail. Therefore, the construction detail No. 4 in Parts 1-9 of Eurocode 3 (details in Table 8.1 of Eurocode 3) with the detail category 140 is selected for the following two reasons: firstly, both methods involve fatigue crack initiation at the cutting edge, as shown in Figure 3; secondly, the direction of crack propagation is perpendicular to the direction of principal stress. The modified nominal stress is defined as the nominal stress multiplied by the stress concentration factor. Therefore, the modified nominal stress around diaphragm cutout can be used to evaluate fatigue life based on the construction detail No. 4. The modified nominal stress ranges in zone A, that considers the lateral probability distributions of five-wheel positions, is defined as follows:  
\[\Delta\sigma_{P}=\sqrt[3]{0.5\times\sigma_{1}^{3}+0.18\times\sigma_{2}^{3}+0.18 \times\sigma_{3}^{3}+0.07\times\sigma_{4}^{3}+0.07\times\sigma_{5}^{3}} \tag{1}\]where \(\sigma_{i}\) is the modified nominal stress range in zone A, which is caused by the fatigue loads with a standard vehicle, and takes into account the local stress magnification in relation to detail geometry, that is not included in the reference curves. The ultimate state equation of the relation between fatigue strength and modified nominal stress range is represented as follows based on a series of S-N-curves, which correspond to typical detail categories. \[\gamma_{\text{FI}}\Delta\sigma_{\text{E2}}=\frac{\Delta\sigma_{\text{C}}}{ \gamma_{\text{Mf}}} \tag{2}\]  
where: \(\gamma_{\text{FI}}\) and \(\gamma_{\text{Mf}}\) are the partial factors for equivalent constant amplitude stress ranges and fatigue strength, respectively, and set as 1.0 for design security and were not taken into consideration in fatigue life prediction in this research; \(\Delta\sigma_{\text{C}}\) is the reference value of the fatigue strength at 2 million cycles, which is 140 for constructional detail No. 4; \(\Delta\sigma_{\text{E2}}\) is the value of modified nominal stress range subjected to 2 million cycles, which be determined as follows:  
\[\Delta\sigma_{\text{E2}}=\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}\Delta \sigma_{P} \tag{3}\]  
\(\lambda_{1}\) is the damage effect factor, which takes the fatigue model III and the length of the influence line into consideration, and the value is 1.85 for diaphragm cutouts. \(\lambda_{2}\) is the traffic flow factor for fatigue model III, which considers the daily traffic volume. \[\lambda_{2}=\left(\frac{0.95\times 365\times p\times N_{y}}{0.5\times 10^{6} \times j}\right)^{\frac{1}{2}} \tag{4}\]  
\(\lambda_{3}\) is the design life effect factor, which is directly related to fatigue life. \[\lambda_{3}=\left(\frac{t}{100}\right)^{\frac{1}{2}} \tag{5}\]  
\(\lambda_{4}\) is the multi-lane effect coefficient, the value is 1 for fatigue model III.