# Current Practices of Railway Ballastless Track Design Methods in China
## 2 Ballastless Track Structure Design Method
### Allowable Stress Method  
The ASM is based on the elastic theory. The calculation principle is that the stress at the most unfavorable position of the structure should be no greater than the allowable stress of material. By adopting a single safety factor K [19], the calculation is convenient. According to the Code for the Design of Railway Tracks (TB10082-2017) [20], the main design process of the allowable stress method is shown in Figure 2.  
According to the specification [20], the loads considered in the ASM include the train load, temperature gradient load, bulk temperature load, concrete shrinkage load (considered as temperature dropping), bridge defection load and subgrade settlement load. 1. Train load  
The loading mode is a single axle with two wheels, and the static wheel load is half of the axle load. For the ASM, according to the specification [20], the train load adopts the design value, calculated as follows:  
The design value of the vertical train load is expressed as:  
\[P_{d}=\alpha\cdot P_{j} \tag{1}\]  
Figure 2: Design process of allowable stress method. where \(P_{d}\) is the vertical design load, \(\alpha\) is the dynamic load coefficient, and \(P_{j}\) is the static wheel weight. And the value of \(\alpha\) is related to the axle weight and speed of the train according to the code [20]. The design value of the lateral train load is as follows:  
\[Q=0.8\cdot P_{j} \tag{2}\]  
According to the code [20], the value of the temperature gradient load is related to the slab thickness. For a thickness of 220 mm, the max temperature gradient is set as 90 \({}^{\circ}\)C/m and \(-45\,^{\circ}\)C/m, and the slab thickness correction coefficient \(\gamma_{t}\) is adopted for others (for the thickness of 200 mm, \(\gamma_{t}=1.05\)). Bulk temperature load  
Generally, the maximum temperature can be determined according to the local temperature conditions. Concrete shrinkage load  
Concrete shrinkage can be equivalently considered as a temperature drop of 10 \({}^{\circ}\)C for the track slab. Bridge defection load  
According to the code [20], the deflection shape is assumed as Formula (5):  
\[y=-\delta\cos\Bigl{(}\frac{\pi x}{L}+\frac{\pi}{2}\Bigr{)}=\delta\sin\Bigl{(} \frac{\pi x}{L}\Bigr{)} \tag{3}\]  
The maximum curvature of the deformation curve is:  
\[\kappa_{\max}=\frac{\pi^{2}}{L^{2}}\delta \tag{4}\]  
where \(\delta\) is the midspan deflection of bridge, and L is the bridge span. Subgrade settlement load  
The shape of the subgrade settlement curve is determined using the formula  
\[y=f_{0}\sin\biggl{(}\frac{\pi x}{l_{0}}\biggr{)} \tag{5}\]  
The maximum curvature of the deformation curve is:  
\[\kappa_{\max}=\frac{\pi^{2}}{l_{0}^{2}}f_{0} \tag{6}\]  
where \(f_{0}\) is the settlement amplitude of subgrade, and \(l_{0}\) is the wavelength of the settlement. Vehicle braking load  
The braking load is less likely to occur, so it is taken as an additional force. The formula is not specified, but some scholars calculate it by applying force equivalent to each fastener with the finite element method [34]. Track structures will generate moment effects under different loads. In Figure 2, \(M_{d}\), \(M_{td}\), \(M_{t}\), \(M_{ntg}\), \(M_{cl}\) and \(M_{zd}\) are design values of moment effects under train load, temperature gradient, bulk temperature (including concrete shrinkage load), bridge defection, subgrade settlement and vehicle braking load. The moment effect can be calculated with the formula in the specification or the finite element method (FEM). Then, various load effects combine and determine the reinforcement of the structure. What is more, for diverse structures, the effect and influence of various loads are different, so different combinations are considered. For the unit structure, the bulk temperature change has little influence on track stress, so only the train load, temperature gradient and bridge defection or subgrade settlement are considered.