# Retrofit Fatigue Cracked Diaphrag Cutouts Using Improved Geometry in Orthotropic Steel Decks
## 3 Geometry Optimization  
### Diaphragm Cutout Stress Concentration  
An finite element analysis (FEA) model was created using the software ABAQUS (V6.13) to calculate the stresses of a steel box girder between two adjacent hangers with a length of 12.0 m, width of 20.4 m, and a height of 3.5 m. The thickness of the deck, U-rib and diaphragm are 16, 8 and 10 mm, respectively. Shell elements are used to model the steel box girder including all components, such as OSDs, diaphragms, webs and so on. The quadrilateral shell element (S4R) is used to enhance computational accuracy and reduce calculation time. Due to the large stress gradients around the diaphragm cutouts, the mesh size of the shell element is refined to 0.2 mm around the cutouts; the element size in other location is 0.3 m. The length-width ratio and distortion of shell elements are checked by the ABAQUS software to ensure mesh quality and accuracy of calculations. The finite element model is shown in Figure 5.  
Figure 4: Repair schemes for diaphragm cutouts: for short cracks using (**a**) Scheme 1; for long cracks using (**b**) Scheme 2, (**c**) Scheme 3 and (**d**) Scheme 4. Some assumptions of the finite element analysis need to be emphasized: (1) the effects of geometric and material nonlinearities are ignored; (2) the hangers on the steel box girder are simplified as rigid supports without considering the effects of elastic and plastic deformations; (3) the effects of residual stresses around cutouts that result from the welded joints between the U-rib and the diaphragm are considered to be small and not considered, since the initiation of the fatigue cracks is located relatively far from the welds; (4) the permanent load or internal residual stresses due to hyperstatic behavior will result in permanent stress in OSD, especially at the weld joints. These permanent stresses will change the R ratio of stress spectrums and hence affect their fatigue lives. In this paper, we ignore the influence of these two effects and just focus on the stress range of different diaphragm cutouts due to standard fatigue vehicle. Except for the above assumptions, the calculations presented in this paper conform to the scope of EN 1993-1-9 in Eurocode 3 (\"Design of steel structures\"). The stress range and fatigue life of cutouts were obtained from the fatigue loading mode III in Eurocode 3. However, the axial weights of middle and rear axles were increased to 140 kN in loading tests and in FEA, to account for the influence of overloading in actual traffic. The contact length and width of the wheel load are 0.3 and 0.7 m in the longitudinal and transversal directions, respectively. Figure 6 presents the relative position between the axles (wheel load locations) and the diaphragm, at which the maximum stress at the edge of diaphragm cutout was obtained by finite element analysis. Therefore, the load case where the middle and rear axles are located on either side of the diaphragm is deemed to be the most unfavorable. The stress contour at the diaphragm cutout under the most unfavorable load case, calculated by FEA, is shown in Figure 7. Two stress concentration areas appear at the edge of the diaphragm cutout, which are marked as zone A and zone B. The maximum Mises stress at zone A is 154.2 MPa, and the maximum value of the principal compressive stress in this zone is minus 155.9 MPa. An on-site inspection of the bridge showed that all the fatigue cracks of the diaphragm were initiated from these zones, and that is in good agreement with the calculated results. Due to the stress concentration in zone A, where the stress is obviously larger than that in other areas, the corresponding large stresses result in fatigue cracks. It should be emphasized that although the principal compressive stress rarely causes fatigue cracks per the theory of linear-elastic fracture mechanics. However, the residual tensile  
Figure 5: Finite element model of steel box girder. Figure 6: The most unfavorable load case and stress flow.