## 1 Introduction  
Concrete tracks, which increase the running stability of high-speed trains, compared with ballast tracks, are widely used in Korea, Germany, China, etc., with the development of high-speed railways. On the other hand, when settlement or cracking occurs in the concrete slab track, the durability of the concrete track deteriorates significantly, and the methods of repair of damaged track are very limited and difficult. A zone where a bridge and embankment or a tunnel and embankment join is called a transition zone. Track support stiffness varies a lot in a short distance in the transition zone. Railway embankment settlement occurs more frequently in the transition zone. An impact load from the dynamic behavior of train operation is mostly generated from differential settlement on the tracked. As the speed of the train increases, such as is the case on a high-speed line, wheel loads on track also increase, due to the impact load in the section where differential settlement occurs in a short distance [1, 2]. [PERSON] and [PERSON] presented analytical transient solutions of one-dimensional finite systems with an abrupt change in foundation stiffness using two methodologies, i.e., a superposition of global modes of vibration and linking together analytical solutions of two separate parts of the structure [3]. The results were used for the determination of critical velocities, and the analysis was conducted with a constant moving force. However, [PERSON] revealed that a vehicle load including suspension systems has a higher accuracy in terms of simulating the impact load than other loading methods, such as a single axial load and a wheel moving load [2]. [PERSON] and [PERSON] investigated a jumping wheel (loss of contact) phenomenon at a transition point using a moving element method [4]. The results showed that the abrupt change in foundation stiffness may trigger jumping wheels in the presence of track irregularities. [PERSON] analyzed principal stresses of the tracked on a settled transition zone using a 3D finite element method (FEM), and the results showed that plastic behaviors of the tracked are caused by shear deformation when the track irregularities and train speeds increase [1]. [PERSON] et al., revealed that when a high-speed train passes bridge-embankment transitions, acceleration and displacement reach a peak at the point 25 m on the tracked from the abutment [5]. At this point, the applied stresses from the wheel load are different from settled and unsettled sections. A numerical analysis approach to analyzing the abovementioned condition requires a realistic simulation for the train and tracked; thus, both the modeling and long analytical periods are difficult to achieve. [PERSON] stated that a track model incorporating all elements of the track, such as fastener, railroad, and sleeper, should be modeled to estimate realistic analysis results [6]. [PERSON] and [PERSON] developed a vehicle-track-subgrade coupling system by utilizing the finite-element method for the transition section model, and the results showed that the transition pattern of the track stiffness has a primary influence on the dynamic behavior of the vehicle and the track, and smoothing of the track stiffness distribution can significantly reduce wheel-rail interaction forces and rail vertical accelerations [7]. [PERSON] conducted a numerical analysis of the transition zone with hanging sleeper, and analyzed wheel-rail contact force for the abrupt change in track stiffness [8]. The analysis consists of a 3D FEM track model and a moving wheelset that simulates the load from one axle of a train. The detrimental effects of hanging sleepers can be reduced by under-sleeper pads. [PERSON] et al., published a numerical model for dynamic loads on the ballast by trains passing the transition, which was then validated against field measurements [9]. The aforementioned numerical analysis was all for ballasted track, and considered the lower tracked as a simple spring or elastic material properties.