## 2 Theoretical Discussion
### Application of Track Stiffness for Calculation of Dynamic Factor  
When applied to ballasted and concrete tracks, the dynamic response of the wheel-rail contact load is different due to the respective track types' stiffness property [13,14]. In this regard, to derive the dynamic response of the track system, stiffness of the track should be a key factor. [PERSON] dynamic factor demonstrates this method, where the equation expresses the standard deviation of dynamic load as an equation related to train speed, variables according to rail and wheel defects, unsprung mass of the vehicle, track support stiffness, and damping of the track. The calculation method is as follows [15]. \[\sigma(\Delta QNS)=0.45\times\frac{V}{100}\times b\times\sqrt{n_{ns}\times K \times\gamma(\epsilon)} \tag{1}\]  
where,  
\(\sigma(\Delta QNS)\): the standard deviation of DAF  
\(V\): train speed  
\(b\): variable of rail and wheel load contact point  
\(m_{ns}\): unsprung mass of the train  
\(K\): track support stiffness  
\(\gamma(\epsilon)\): rail damping  
Among the variables, track support stiffness (\(K\)) is an indicator of the track structure's bearing capacity against dynamic loading of train wheel loads. This factor takes into consideration the structural characteristics and elastic range of the track system [15,16]. Track support stiffness can be understood as the vertical stiffness of the track structure, derived by the comprehensive spring coefficient from the individual components that comprise the entire track structure. The following equation is implemented where the spring coefficient \(k_{s}\) is derived by calculating the spring coefficient (stiffness) of the individual components/layers of the ballasted track structure) and the spring coefficient \(k_{s}\) is used to calculated the TSS of the structure [17]  
\[k_{bs}=\frac{1}{\frac{1}{k_{p}}+\frac{1}{k_{d}}+\frac{1}{k_{b}}+\frac{1}{k_{ sg}}} \tag{2}\]  
where,  
\(k_{p}\): the stiffness of the rail pad (kN/mm),  
\(k_{sl}\): the stiffness of the sleeper (due to the compressibility of wood in the rail-seat region and sleeper bending) (kN/mm),  
\(k_{b}\): the stiffness of the ballast layer (kN/mm),  
\(k_{sg}\): the stiffness of the subgrade (kN/mm), and \(k_{\text{bs}}\): the stiffness of the ballasted track structure (kN/mm)  
\[k_{\text{bss}}=\frac{1}{\frac{1}{k_{p}}+\frac{1}{k_{r}}+\frac{1}{k_{d}}+\frac{1} {k_{\text{slab}}}+\frac{1}{k_{\text{sg}}}} \tag{3}\]  
where,  
\(k_{p}\): the stiffness of the pad (kN/mm),  
\(k_{r}\): the stiffness of the resilience (if applicable) (kN/mm),  
\(k_{\text{sl}}\): the stiffness of the sleeper (if applicable) (kN/mm),  
\(k_{\text{slab}}\): the stiffness of the ballast-less slab (kN/mm),  
\(k_{\text{sg}}\): the stiffness of the subgrade (kN/mm),  
\(k_{\text{bss}}\): the stiffness of the ballast-less slab track structure (kN/mm)  
\[K_{t}=\sqrt[4]{\frac{64 EI}{d^{3}}}K_{\text{s}}^{3} \tag{4}\]  
E: modulus of elasticity (GPa),  
I: moment of inertia (mm\({}^{4}\)),  
d: distance of the sleepers (mm),  
\(k_{\text{s}}\): stiffness of the track structure (kN/mm),  
\(k_{t}\): track support stiffness (kN/mm). For the trackside measurement of TSS, [PERSON]'s law can be referred to in order to derive the track support stiffness of the track structure, whereby the track is considered as a beam structure, and the track support stiffness is derived using the variables of maximum dynamic wheel load during trackside measurement and the maximum rail displacement [18];  
\[K_{t}=\frac{Q_{dyn}}{\delta_{max}} \tag{5}\]  
where,  
\(K_{t}\): is the trackside measurement based track support stiffness (total spring constant) (kN/mm);  
\(P_{max}\): is dynamic wheel load (maximum value derived from trackside measurement) (kN);  
\(\delta_{max}\): is rail vertical displacement (maximum value derived from trackside measurement (mm). The dynamic factor calculation method of KR C-14030 is used when designing the track. This is based on [PERSON]'s equation that employs train vehicle speed and track quality as variables.