# Load Transfer Efficiency Based on Structural Deflection Assessment of the Precast Floating Track
## 4 Numerical Analysis  
In railway application, according to the Zimmermann method which is the well-known \"beam on elastic foundation\", the rail is assumed as a continuous beam supported by the elastic foundation system composed by fastening system, ballast, sub-ballast-mat as well as sub-soil. In this study, we focused on the displacement of the structures under the static loads to determine the load transfer efficiency of precast floating slab track. To model this type of precast floating slab track, based on this method, we considered the rail as a continuous beam mounted on the panels by fastening system and the panels (discontinuous slabs) was attached with the anti-vibration device as the elastic component.  
According to the beam on elastic foundation theory, the deflection and moment of the beam under the concentrated wheel load shown in Figure 8 can be calculated as following formulas:  
\[w(x)=\ \frac{QL^{3}}{8 EI}\eta(x) \tag{1}\]  
\[M(x)=\ \frac{QL}{4}\mu(x) \tag{2}\]  
where: \(L\): characteristic length = \(\sqrt[4]{\frac{AEI\mu}{K_{d}}}\) (m); and \(k_{d}\): stiffness coefficient of discrete support and \(\frac{1}{k_{d}}=\sum_{i}^{1}\frac{1}{k_{i}}\);  
\(a\): spacing between centers of discrete supports (m);  
\(Q\): wheel load (N) = 0.5 P (with P is axle load)  
\(EI\): bending stiffness of beam (N/m\({}^{2}\));  
and, two influence factors are:  
\[\eta(x)=\ e^{-x/L}\Big{[}cos\frac{x}{L}+sin\frac{x}{L}\Big{]}\ x\geq 0 \tag{3}\]  
\[\mu(x)=\ e^{-x/L}\Big{[}cos\frac{x}{L}-sin\frac{x}{L}\Big{]}\ x\geq 0 \tag{4}\]  
Figure 7: Locations of sensors to evaluate the performance of the three precast panels.  
This type of precast floating slab track was simulated as the double beam model with discrete support so the finite element method can be used to evaluate the structural behavior of the track [1]. In this paper, we used MIDAS CIVIL 2019, a finite element program to model the precast floating slab track. The continuous rail was modeled by using the profile of KR 60 rail, considered as a continuous beam and the standard gauge (1435 mm) was applied. The panel was modeled by two concrete slabs connected with each other by crossbeams using the beam element and the distance between each panel is 75 mm. The rail and concrete slab were connected by the elastic fastening system (system 300-1, KR type) with the vertical stiffness was 28.7 kN/mm. Six anti-vibration devices were set up at the bottom of each panel as the spring device (elastic link element), the vertical stiffness of each device was 22.5 kN/mm. The structure system analysis model and specifications are shown in Figure 9 and Table 3, respectively.  
Figure 8: Beam on elastic foundation model.  
Figure 9: Modeling of the precast floating slab; (**a**) overview, and (**b**) front view.  
Figure 9 shows the concept of three panels (approximately 15 m length) simulated as the same dimensions of the actual specimens with the continuously rail mounted on by the fastening device. This new type of precast floating slab track has no joint to connect the panels, so the load can be distributed and transmitted directly through the rails.