# Retrofit Fatigue Cracked Diaphrag Cutouts Using Improved Geometry in Orthotropic Steel Decks
## 3 Geometry Optimization
### Optimization of Diaphragm Cutouts with Fatigue Cracks  
The cutout geometry can be represented by three parameters \(r\), \(l\) and \(\beta\), as shown in Figure 10, where, \(r\) is the radius of the upper arc segment, \(l\) is the length of the upper line segment that connects the upper arc segment to the U-rib, and \(\beta\) is the inclined angle between the upper line segment and the \(x\)-axis; the inclined angle is positive when it is in the first quadrant, and is negative when it is in the fourth quadrant. The entire geometrical shape of the cutout is also affected by the geometries of several other parts, such as the lower arc segment and the lower line segment. However, they have little effect on the stresses at the critical locations of the diaphragm.  
The finite element model depicted in Figure 5 and the most unfavorable load case were used to calculate and analyze the concentrated stress in the diaphragm cutout area, with different parameters \(r\), \(l\) and \(\beta\). Zone A is located at the edge of the diaphragm cutout, which is a planar structure, and hence the direction of the major principal stress is tangential to the arc, while the other two principal stresses are very small. Therefore, geometry optimization was carried out based on the major principal stresses reported in this paper.  
When the straight-line length \(l=10\) mm and the included angle \(\beta=0^{\circ}\), the arc radius r increases from 10 to 40 mm, and the major principal stresses at zone A decrease with an increase in the diaphragm cutout radius, as shown in Figure 11a. The principal stress increased obviously when the arc radius is small, as radius increasing, the reduction degree gradually decreased. Due to the increase in the arc radius, the stress concentration effect is weakened, which is beneficial in improving the fatigue life of the diaphragm cutout. Therefore, the larger the arc radius, the better it is for reducing stress concentrations, and it is recommended that the arc radius should be not less than 30 mm.  
Figure 10: Shape parameters of a diaphragm cutout.  
Figure 9: Stress flow analysis around a cutout at the U-rib to diaphragm connection: (**a**) vectors of principal stresses and (**b**) stress flow and load path.  
When the arc radius \(r=30\) mm and the included angle \(\beta=0^{\circ}\), an increase in the straight segment 1 from 10 to 40 mm causes a linear increase in the major principal stresses of zone A, as shown in Figure 11b. When the radius of the cutout arc is maintained constant, the hole of the cutout becomes larger due to the elongation of the straight section, which in turn obstructs the diffusion of the stress flow near the cutout and leads to an increase in the major principal stresses. Therefore, it is not suitable to set the straight-line segment at the upper cutouts, and smaller diaphragm cutouts are more favorable.  
When the straight line \(l=20\) mm and the arc radius \(r=30\) mm, the increase in peak stresses are accelerated as the inclined angle of the line segment increases from \(-60^{\circ}\) to 0 as shown in Figure 11c. Since the radius of the diaphragm cutout is constant, the stress concentration effects are approximately equal, and the increase of the peak stresses are mainly caused by the obstruction of stress flows due to the different included angles. The direction of transmission of the stress flows are inclined downwards, as shown in Figure 9b. Therefore, the included angle of the straight segment should be less than \(0^{\circ}\).  
In conclusion, the geometry optimization principles of diaphragm cutouts can be summarized as follows: (1) the radius r of diaphragm cutouts should be large; (2) the geometrical size of diaphragm cutouts should be small; and (3) the included angle of U-ribs should be small (a negative value is considered small). In other words, the shape of the cutouts should avoid obstructing the stress flows.