# Buckling Analysis of Interspersed Railway Tracks
## 2 Methodology
### Linear Eigenvalue Analysis  
In this study, linear buckling analysis was used to predict the buckling temperature. Firstly, boundary conditions were considered to study the lengths of the buckling and adjoining zones. After preliminary study on the buckling zone, it was found that the boundary condition has a slight effect on the buckling temperature but plays a significant role in the buckling shape. To reduce the time consumption, only buckling temperature was studied using linear eigenvalue analysis.  
Importantly, the buckling regime is mostly in the range between the maximum and minimum buckling temperatures, as shown in Figure 4. It is noted that the maximum buckling temperature (T\({}_{\text{Bmax}}\)) is the inflection path, where the pre-buckling stage becomes the post-buckling stage, also known as the \"bifurcation or critical temperature\". The minimum buckling temperature (T\({}_{\text{Bmin}}\)) is the lower bound which can buckle the track if sufficient energy is supplied. It can also be defined as a safe temperature since the track cannot buckle if it experiences heat below this temperature. It should be noted that only the bifurcation temperature, where the track suddenly buckles and becomes unstable, was evaluated in this study.  
It is important to note that eigen buckling analysis predicts the presence of the bifurcation point at which the primary load-deflection path is bifurcated by a secondary load-deflection path, as shown in Figure 5, and it can occur at more than one equilibrium position at this point [51; 55; 56]. It is noted that, after buckling, the post-buckling state does not follow the primary path. In this stage, the secondary slope can be either positive (post-buckling strength) or negative (simply collapse). The governing equation used for calculating the buckling temperature and corresponding buckling shapes is shown in Equation (1).  
\[\left|K+\lambda K_{g}\right|=0\]  
Figure 4: Buckling path.  
This equation is equivalent to the eigenvalue solution as shown in Equation (2):  
\[Kx=-\lambda K_{g}x \tag{2}\]  
where \(K\) is the global stiffness matrix, \(x\) is buckling mode vectors, \(\lambda\) is the buckling load factor (BLF), and \(K_{g}\) is a global geometric stiffness matrix known as the initial stress stiffness matrix depending on the stress level of the element. It is noted that the element force vector at each state changes the geometric stiffness of the element.  
It is important to note that the buckling load factor (BLF) is an indicator of the safety factor (FOS) against buckling or the ratio of the buckling load to the currently applied loads. By using Equation (2), the BLFs are calculated and then multiplied by the applied temperature to get the buckling temperature. In this study, 100 \({}^{\circ}\)C was applied to the whole system. It should be noted that buckling is predicted when the applied thermal loads are higher than the estimated critical loads (BLF \(<\) 1). On the other hand, the track is not buckled under the applied temperature when the BLF is beyond 1. However, linear eigenvalue analysis can still calculate the buckling temperature by extrapolating to reach the buckling temperature, even though the possible buckling temperature is higher than the applied temperature.