## 3 Numerical Analysis  
### Modeling  
Previous research, and damage evolution and fatigue models for various engineering materials, were reviewed [22]. In this study, the structure can no longer be used when it exhibits plastic behavior, the numerical analysis of the structure assumes linear elastic behavior. When heaving and settlement occurred in subway box structure, its safety was confirmed through numerical analysis. Using the continuous elastic foundation beam analysis employing the displacement measurement for each location, obtained based on the in situ measurements, loads and boundary conditions that were employed in the 3D finite element analysis model were applied. The analysis of beams in elastic ground was assumed to be proportional to the degree of deformation rather than the reaction force of the elastic ground. In 1867, [PERSON] proposed the continuous elastic foundation beam theory [23]. It is a technique used to examine the behavior of flexible structural elements in contact with supporting and bearing soils as an infinite continuum by idealization employing a simplified continuum model, as shown in Figure 4.  
Figure 4 illustrates the assumption of a railway track modeled with an infinite Euler-Bernoulli beam, which is continuously supported in the longitudinal direction. The deflection \(\omega(x)\) of the rail can be computed using the differential equation in Equation (1) using the following two track parameters. In Equation (1), q(\(x\)) is the reaction force of the elastic ground according to deflection \(\omega(x)\) at any point of the beam. Furthermore, the subgrade reaction modulus in the vertical direction can be computed using Equations (2) and (3), respectively [24]. \[EI\frac{d^{4}\omega(x)}{dx^{4}}+\mathrm{k}\omega(x)=\mathrm{q}(x), \tag{1}\]  
\[K_{v}=K_{v0}\left(\frac{B_{v}}{0.3}\right)^{\frac{-3}{4}}, \tag{2}\]  
Figure 4: Beam on continuous elastic foundation. Figure 3: Comparison between optical leveling and visual inspection results for each section. \[K_{v0}=\frac{1}{0.3}\times\alpha\times E_{0}, \tag{3}\]  
where q(\(x\)) is the distribution load on the rail (N/m), _EI_ is the flexural rigidity (k/m\({}^{2}\)), k is the foundation rigidity (N/m\({}^{2}\)), \(d\) is the deformation (m), \(\omega(x)\) is the deflection (m), \(K_{v}\) is the coefficient of the vertical subgrade reaction (kN/m\({}^{3}\)), \(K_{v0}\) is the initial coefficient of the vertical subgrade reaction (kN/m\({}^{3}\)), \(B_{v}\) is the section modulus (m), \(E_{0}\) is the deformation modulus (kN/m\({}^{2}\)), and \(\alpha\) is the correction factor of the deformation modulus (constant). The in situ measurement results were applied to Equations (1)-(3) to compute the subgrade reaction modulus of the structure, as presented in Table 1 and Figure 5. Figure 5 shows the results of the subgrade reaction modulus calculated by the displacement function using the leveling results and structural analysis. The maximum heating section occurred in the 20-30 m range, and the maximum settlement section occurred in the 60-70 m range. Additionally, the deformation function of the structure and the subgrade reaction modulus for each displacement measurement point were applied as the load and boundary conditions in the 3D numerical analysis model. As shown in Table 1, the subgrade reaction modulus due to settlement and heating differed depending on the location. Additionally, the subgrade reaction modulus in the maximum settlement section was reduced by approximately 95% compared to that in the maximum heating section.