## 3 Finite Element Analysis of a Railway Track Slab with a Gap  
### Numerical Modeling of Track Slab  
In this study, ABAQUS software [21] was used for simulating a nondestructive test (NDT) on a concrete slab representing a high-speed railway track ballast. Stress wave propagation was simulated through a transient response analysis on a plane strain finite element model by applying an imaginary impact load to a model concrete slab surface. A concrete slab was modeled to be constructed on a homogeneous half space. Two models of a railway track slab were assumed for simulation comparison purposes. One was modeled as a section of railway track slab with a gap (diameter D = 1.0 m), and the other was modeled as a section of railway track slab without a gap, as shown in Figure 1. The diameter of the gap was selected from the results of a previous study that was focused on investigation of the stability of tracked foundation with different sizes of gap loaded by railway car on top of a concrete slab track [22]. From the developed stability chart, a sensitive parameter consisting of gap depth z = 0 m, gap diameter D = 1.0 m, and soil modulus E = 40 MPa was chosen in this study since this sensitive parameter is located on a threshold curve in the chart where the tracked foundation under the slab can be endangered. In addition, in a practical point of view for the authority (KORAIL), a gap smaller than 1.0 m diameter of the gap is little bit more difficult to be detected using the spectrum generated by rail car wheel load since the concrete slab is severely thick (1.0 m) and its bending stiffness EI (E=concrete elastic modulus, I= moment of inertia) in the longitudinal direction is big enough [22]. The finite element model was 10 m high, 9 m wide, and included air elements above the slab. For simple modeling, the elastic modulus of the soil was assumed as 40 MPa, the concrete was 3 GPa, and the air was 0.1 MPa, which was obtained from the ideal gas equation. The depth of the soil below the track slab was 5.0 m. The thickness of the concrete slab was 1.0 m, and this simulated the real thickness of a high-speed railway track slab. The air zone above the track was modeled as 3.0 m high. An imaginary impact load was applied to the concrete surface for \(3.0\times 10^{-5}\) s, and it was simulated as being a half sine wave and its peak load was 250 N. The simulated maximum load was obtained  
Figure 1: Finite element modeling of railway track slab. (**a**) Without gap. (**b**) With gap. from the impact of a sledgehammer as a general impact echo (IE) test applied onto the concrete surface in the field. The duration time of the impact \(t_{c}\) was calculated by considering the following equation:  
\[t_{c}=5.97\big{[}\rho_{s}(\delta_{s}+\delta_{p})\big{]}^{0.4}(R_{sphere}/H^{0.1}) \tag{5}\]  
where \(\delta_{p}=(1-{\
u_{p}}^{2})/E_{p}\), \(\delta_{s}=(1-{\
u_{s}}^{2})/E_{s}\), \(\rho_{s}\) is the density of the imaginary steel ball or sledge hammer head, \(R_{sphere}\) is the radius of the steel ball, \(\
u_{s}\) is the Poisson's ratio of the steel ball, and \(E_{p}\) is the elastic modulus of the steel ball. For the time of impact loading (\(t_{c}\)), impact energy equivalent to the energy level of the sledge hammer loading applied to the slab surface was used to compute it by considering an imaginary falling height of an equivalent small steel ball with the same weight of the sledge hammer head. [PERSON] and [PERSON] [23] proposed that the maximum frequency of an impact test could be represented as:  
\[f_{\text{max}}=1.25/t_{c} \tag{6}\]  
Therefore, the minimum wavelength can be calculated by the following basic equation:  
\[\lambda=V_{c}/f_{\text{max}} \tag{7}\]  
where \(V_{c}\) = the compressive P-wave velocity of the material to be tested. For the characteristics of force-time function and frequency distribution, [PERSON] [20] studied the distribution of the frequency according to the diameter of various steel balls and the force-time function.