# Comparison of Fatigue Performances Based on Shape Change of Rail Fastening Spring
## 3 Comparison of Fatigue Performance Based on Fastening Spring Shapes
### Overall Spring Width  
The effect of the overall fastening spring width on the fatigue performance was evaluated with five different spring widths, including the original one. Figure 11 shows the perspective view of the spring width change. The widths were changed from \(-20\%\) to \(20\%\) of the original one with \(10\%\) increments. Figure 12 depicts the fastening force vs. displacement diagram for various spring widths. The required vertical displacements corresponding to the fastening force (9 kN) could be reduced as the width becomes smaller.  
Figure 12 compares the clamping force at the torque location with the fastening force at the rail-to-spring boundary of the interface. By equilibrium, the clamping force of the torque can be resisted at the two boundary locations of the rail-to-spring and guide plate-to-spring interfaces. In Figure 12, the ratio of the clamping force at the torque to the rail fastening force was approximately \(61\%\) for the \(+20\%\) model. This ratio was approximately the same for the other models.  
Figure 13 illustrates the maximum principal stresses during fastening when varying the spring width, while (b) shows the range of the maximum principal stress due to the vertical deflection caused by train use. Both the maximum principal stress at the initial fastening and the stress range caused by the vertical deflection tended to decrease as  
Figure 8: Perspective view of the change in the spring arm height.  
Figure 10: Stress variation with the spring arm height: (**a**) maximum principal stress; (**b**) maximum principal stress range.  
Figure 9: Fastening force vs. displacement diagram varying the spring arm height.  
the width increased. Although unclear, the curvature that can be related to the stress concentration on the critical sections of the original shape appeared higher than that of the one with a width increment. This curvature effect on the maximum principal stress might be more significant than the effect of the internal force increase at the critical sections with the width increment.  
Figure 14 shows a comparison of the fatigue performances varying with the overall spring width based on the analysis results in Figure 13. The comparison in Figure 13a is based on the modified Goodman criterion, which can be more conservative than the Gerber criterion. The safety margin (%) is defined herein as an index for comparing the fatigue performances of each model. It is the ratio of the remaining stress amplitude to the allowable stress amplitude in the modified Goodman diagram. Therefore, when the safety margin is high, the fatigue performance is enhanced. In Figure 14, the spring width  
Figure 11: Perspective view of the change in the spring width: (**a**) front; (**b**) plan views.  
Figure 12: Force vs. displacement diagram: (**a**) fastening force vs. displacement when varying the spring width; (**b**) comparison of the clamping force at the torque with the rail fastening force for the +20% model.  
Figure 13: Stress variation with the spring width: (**a**) maximum principal stress; (**b**) maximum principal stress range.  
modifications of \(-20\%\), \(-10\%\), \(0\%\) (original shape), \(+10\%\), and \(+20\%\) yielded safety margins of 64%, 73%, 75%, 80%, and 82%, respectively. The mean stress and stress amplitude decreased as the spring width increased. Based on the modified Goodman criterion, a larger safety margin can be achieved with an increased spring width.