# Comparison of Fatigue Performances Based on Shape Change of Rail Fastening Spring
## 3 Comparison of Fatigue Performance Based on Fastening Spring Shapes  
In order to limit various physical responses, such as axial force and displacement generated by the track-bridge interaction in Korea, the European design standard UIC-774-3(R) [20] is applied to define and apply KRC-08080 [21]. According to KRC-08080, the  
Figure 4: Failure modes of KR-type fastening spring from fatigue testing.  
Figure 3: Maximum principal stress contours: (**a**) initial fastening (Step 1); (**b**) vertical displacement (Step 2).  
Figure 2: Finite element analysis modeling: (**a**) model configuration; (**b**) material model.  
longitudinal load-displacement diagram of the track was modeled, and the main variables that determine the characteristics of the longitudinal resistance of the limit displacement track are defined as the limit displacement \(u_{0}\) and the longitudinal resistance \(F_{0}\). The longitudinal resistance of the rail fastening system was determined as follows by the frictional resistance caused by the initial fastening force:  
\[R_{L}=\mu F_{0} \tag{1}\]  
where \(R_{L}\) is the longitudinal resistance and \(\mu\) is the static friction coefficient between the rail and rail pad.  
According to KRS TR0014-13R [22], the standard value of the longitudinal resistance of the rail fastening system should be \(\geq\)7 kN for type C (speed of general railways \(<\)250 km/h) and \(\geq\)9 kN for type D (speed of high-speed railways \(>\)250 km/h). In the case of the rail fastening system for a slab track, the static friction coefficient between the rail and the rail pad was assumed to be 0.5 [22]. Therefore, the initial fastening force must be greater than 18 kN to secure a minimum longitudinal resistance of 9 kN for type D of our study of interest. The fastening springs were installed on both sides of the rail, resulting in a clamping force of \(\geq\)9 kN for a single fastening spring. Herein, a fastening force of 9 kN was set as the fastening requirement.  
The fatigue performance of the fastening spring was determined by the fracture criterion defined using the stress amplitude, mean stress, and material strengths under constant amplitude loading [23]. [PERSON] et al. [12] stated that the measured stress amplitude and vertical displacement in the rail fastening spring during service showed almost consistent stress and displacement variations for each train operation. Among several fracture criteria, the modified Goodman [19] and Gerber [24] criteria are widely used to represent the lower and upper bounds, respectively, of fatigue performance [18].  
\[\text{Modified Goodman}:\ \frac{\sigma_{m}}{S_{U}}+\frac{\sigma_{a}}{S_{E}}\leq 1 \tag{2}\]  
\[\text{Gerber}:\ \left(\frac{\sigma_{m}}{S_{U}}\right)^{2}+\frac{\sigma_{a}}{S_{E}} \leq 1 \tag{3}\]  
where \(S_{U}\) and \(S_{E}\) are the tensile strength and endurance limit of the material, respectively. The yield and tensile strengths of the fastening spring used herein were 1377 and 1509 MPa, respectively. The endurance limit, \(S_{E}\), which is the fatigue strength of a metallic material under stress reversal, generally ranged from 35% to 60% of the tensile strength of the material [18; 25]. The endurance limit was conservatively assumed to be 35% of the tensile strength of the spring steel.