## 2 Theoretical Approach  
Figure 3 shows the axial stress distribution in a CWR on a bridge when the temperature rises. The stress is generated in the rail due to temperature change and ballast resistance, and the immovable zone occurs in the center of the CWR, in which the extension or contraction of the CWR is restricted. When a bridge is located in the immovable zone, additional axial stress in a CWR is generated due to TBI, as shown in Figure 3, because, unlike the CWR, the length of the bridge can be changed by temperature variations [8]. If the temperature rises excessively, buckling of the CWR occurs when the summation of the axial stresses in the CWR reaches the critical buckling stress, as illustrated in Figure 3.  
Based on Figure 3, the allowable additional axial stress of the CWR track on the bridge region, \(f_{all}\), can be expressed as:  
\[f_{all}\ =\ f_{b,min}-f_{T}\ =\ \alpha_{R}E_{R}T_{b,min}-\alpha_{R}E_{R}\Delta T_{R} \tag{1}\]  
Figure 3: Axial force distribution of a CWR track on a bridge. Figure 2: Application example of new girder-sleeper fastener device for a CWR track on an open-deck steel plate girder bridge in Korea. where \(f_{b,min}\) is the minimum buckling stress of the CWR track, \(f_{T}\) is the stress due to the temperature change, \(\Delta T_{R}\). \(\alpha_{R}\) and \(E_{R}\) are the thermal expansion coefficient (=1.14 \(\times\) 10\({}^{-5}\)/\({}^{\circ}\)C) and Young's modulus (=210,000 MPa) of the CWR, respectively. The concept of additional axial stress of the CWR in compression is shown in Figure 4. Generally, the relationship between the axial stress and lateral displacement of the rail for CWR track buckling shows a snap-through shape, and the critical buckling stress of the CWR track is defined as \(f_{b,min}\) (refer to Figure 4). Thus, the additional axial stress of the CWR can be obtained by subtracting the stress due to temperature change, \(f_{T}\), from \(f_{b,min}\). In UIC 774-3R [9], the allowable additional axial compressive stress is specified as 72 MPa for a ballasted track with a minimum curve radius of 1500 m and a consolidated deep ballast (more than 30 cm of ballast height). Similar to the UIC code, KR C-08080 [10], a Korean code for track design, gives allowable additional axial stress (compression) for a ballasted track, as shown in Table 1. When the curve radius is greater than 1500 m, the allowable additional axial stress is the same as that of UIC code. In addition, it can be seen that KR C-08080 provides more detailed allowable additional axial stress criteria for various curve radii. KR C-14050 [11] provides the empirical formula to calculate the minimum buckling stress, \(f_{b,min}\), and \(f_{b,min}\) can be obtained as the minimum value of:  
\[f_{b,min2}\ =\ \frac{10.6I_{R}^{0.388}g^{0.521}}{A_{R}}\times 10\ \text{For}\ R\geq R_{0} \tag{2}\]  
and  
\[f_{b,min1}\ =\ \frac{11.6I_{R}^{0.374}g^{0.534}-aI_{R}^{b}g_{0}{}^{c}/R}{A_{R} }\times 10\ \text{For}\ R<R_{0}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Curve Radius, \(R\) (m)** & **Allowable Additional Stress (Compression, MPa)** \\ \hline \(R\geq 1500\) & 72 \\ \(R\geq 700\) & 58 \\ \(R\geq 600\) & 54 \\ \(R\geq 300\) & 27 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Allowable additional stress standard according to a ballasted track (KR code). Figure 4: Concept of additional axial stress (compression) calculation. where  
\[R_{0}\ =\ \frac{aI_{R}^{\ b}g_{0}{}^{c}}{11.6I_{R}^{\ 0.374}g_{0}{}^{0.534}-10.6I_{R}^{ \ 0.388}g_{0}{}^{0.521}}\]  
In Equation (2), \(f_{b,min1}\) and \(f_{b,min2}\) (MPa) are the minimum buckling stress of a CWR track with symmetric and anti-symmetric modes, respectively, as shown in Figure 5. \(I_{R}\) is the bending rigidity of the rail about a weak axis (cm\({}^{4}\)), \(g_{0}\) is the peak ballast lateral resistance per rail (N/cm/rail), \(A_{R}\) is the area of the rail (cm\({}^{2}\)), and \(R\) is the curve radius (m). Constant values of \(a\), \(b\), and \(c\) are 109, 1.04, and \(-0.186\), respectively for a 60-kg rail.