Evaluation of Structural Stresses of Mountain-Embedded Railway Systems

Abstract

Mountain railways are constructed on terrains with steep gradients and operate under unique conditions as they climb slopes at low speeds using rack rails. Recently, mountain railway systems in the form of embedded rail systems (ERSs) have been developed to minimize environmental damage caused by the construction of new roads. However, the use of ERS in mountain railway tracks is still limited, and research on their safety and behavior is lacking. Therefore, this study was conducted to evaluate the safety profile and behavioral responses of mountain-ERS under a steep gradient condition of 180‰. The effects of using high-strength concrete materials for precast concrete panels were investigated in this study, and the safety profile and behavioral response of mountain-ERS based on changes in the gradient and after adjusting the thickness of the subgrade of mountain-ERS was assessed. Under load conditions, the maximum tensile stress in the concrete elements did not exceed the tensile strength of the concrete. In the structural behavioral analysis, the patterns of stress variation were analyzed by applying stress to the concrete elements. The safety assessment and behavioral analysis results obtained in this study are considered valuable foundational research data for analytical studies on mountain-ERSs.

1. Introduction

Railway systems are crucial transportation infrastructures that provide considerable social benefits through the movement of goods and passengers. Mountain railway tracks, usually designed for tourism, operate under unique conditions in which rack rails traverse sloping terrain at lower speeds. Furthermore, it is important to ensure safety and enhance the strength and rigidity of mountain railway track systems because additional loads are applied to them. In the case of mountain railways, the gear-shaped pinion at the center of the train engages with the rack rails of the railway track, thus enabling the train to start and brake. To enable a train to ascend steep slopes, the propulsion force is transmitted to the mountain railway track system through rack rails. Conversely, safe and controlled travel is ensured during descent by controlling the mountain train speeds using these rack rails.

A mountain railway system for tourism was recently developed in the Republic of Korea. Most of the roads, having been designed for automobiles, are characterized by steep slopes and sharp curves. Existing automobile roads are adapted in constructing mountain railways to minimize their environmental impacts. For this purpose, an embedded rail system (ERS) for mountain railway tracks is currently being developed [1,2,3]. As shown in Figure 1, an ERS is a composite concrete railway track system in which rails are affixed to a concrete track using synthetic resin, and rack rails are installed at the center of the concrete panel. One of its key advantages is its ability to be integrated into existing roadways, thereby reducing environmental damage compared to conventional railway track systems. Fixed rails ensure safety and stability while reducing noise and vibrations owing to the absence of direct contact between the rails and concrete panels. The materials and specifications of the structural components of the mountain-ERS are listed in

Table 1. Material properties of components of mountain railway track model.

No	Component	Materials	Compressive Strength (Concrete) /Tensile Strength (Steel)	Poisson’s Ratio
1	50-kg rails	steel	800 MPa	0.3
2	precast concrete panel	concrete	45 MPa	0.18
3	mortar filler layer (bedding layer)	mortar	45 MPa	0.18
4	subgrade layer	concrete	24 MPa	0.18
5	shear anchor	steel	400 MPa	0.3
6	synthetic resin	synthetic resin	23 MPa	0.33–0.35

The numbering corresponds to Figure 1.

.

Major railways are primarily designed and constructed as ballasted tracks. Conversely, concrete tracks have been predominantly applied to high-speed train tracks. ERSs, which are a specific type of concrete track, have been utilized in urban areas, often running alongside roadways, to accommodate vehicular traffic. Consequently, research in this domain has predominantly focused on ensuring operational safety and providing passenger comfort. So far, research related to ERSs has included analyses of the various factors, specifically, wheel–rail impact in an embedded rail system [4], the dynamic behavior between the railway track system and vehicles [5,6], and the optimized cross-section of the ERS track and its vertical stiffness [7]. In addition, the attachment performance and modeling methods used for the structural analysis of synthetic resin materials for fastening rails to concrete panels have been studied [8,9].

However, only a few cases exist wherein ERSs have been applied to mountain railway systems, and research on the safety profile and behavioral responses of ERS on installed rack rails has been limited. Mountain-ERSs, with respect to this project, are under-researched [1,2,3,7]. A study has been conducted on the application of precast concrete panels in high-gradient mountain railways [1]. In this study, the prerequisites for applying ERS to mountain railways were investigated. Rails appropriate for mountain-ERS were considered, the material properties of the components were analyzed, and the conditions for determining the shape of the ERS track were evaluated. Second, in [2], a study was conducted on the design specifications for mountain railway alignment in Korea, considering the characteristics of Korean mountain trains and car roads in mountainous areas. In this study, the cant, radius, and length of the planar, as well as transitional, longitudinal, and slack curves, were investigated based on the performance and specifications of the Korean mountain railway. In [3], the applicability of 50-kg rails to sharp curves on mountain railway tracks was evaluated. In this study, the material characteristics and residual stress of a 50-kg rail were evaluated and quantified based on experimental and finite element method analyses of rails according to the bending process.

A review of previous studies reveals that researchers on ERS have mainly focused on the analysis of the dynamic behavior of ERS and the interactions between wheel and rail systems. In addition, the elastic poured compounds (synthetic resin materials) of ERS, with respect to the attachment performance of the rail and concrete track, have been researched in experiments [9]. Limited research has been conducted on the application of ERS to mountain railways, and no research has been conducted on the safety profile and behavioral responses of ERSs applied to mountain railways.

Therefore, in this study, safety evaluations and behavioral analyses of mountain-ERSs (with a maximum gradient of 180‰) were conducted. A three-dimensional structural analysis model of mountain-ERS was developed. Service loads with factors suggested in the DIN (Deutsches Institut für Normung, (Berlin, Germany)) Technical Report 101 [10] were considered along with an adequate boundary condition reflecting the site conditions in which the mountain-ERS would be installed. Safety reviews of these structures were conducted by investigating the maximum tensile stress and comparing the tensile strengths of concrete materials. The characteristics of the behavioral responses of mountain-ERSs with high-strength precast concrete panels, safety reviews, and behavioral analyses of the systems (subject to gradient and subgrade layer thickness changes) were analyzed.

2. Simulation Model for the Mountain-ERS Track

2.1. Mountain-ERS Track

The mountain-ERS track was designed for construction on a steep slope of 180‰. As shown in Figure 2, the track was composed of 50-kg rails, a precast concrete panel, longitudinal and transverse reinforcing bars inside the panel, a synthetic resin that fastened the rails, a subgrade layer, shear anchors resisting slippage, and a bedding layer between the concrete panel and concrete subgrade layer with infilled mortar. The bedding layer allowed the precast concrete panel to be stably placed on the subgrade and played a role in fixing the concrete panel to the subgrade layer. The subgrade layer supported the entire structural system and transferred the loads to the ground. A rack rail was installed in the middle of the concrete panel to allow mountain trains to ascend or descend slopes.

The length and width of a unit of the straight railway track were 3500 and 1700 mm, respectively. The heights of the precast concrete panel, bedding layer, and subgrade layer were 290 mm, 50 mm, and 200 mm, respectively. As shown in Figure 3, the height and width of the train were 3785 mm and 2400 mm, respectively. The mountain train consisted of three connected cars with an interval distance of 7000 mm between them, and the distance between the front and rear wheels was 1650 mm.

2.2. Finite Element Analysis Model for the Mountain-ERS Track

A three-dimensional structural analysis model for railway tracks was developed using Abaqus 2022, a commonly used structural analysis program. As shown in Figure 4, the 50-kg rails, precast concrete panel, reinforcement bars inside the concrete panel, bedding layer, and concrete subgrade layer were modeled. The precast concrete panel, bedding layer, and subgrade layer were modeled using solid elements (C3D8R). Regarding the steel material components, transverse, and longitudinal reinforcing bars (arranged inside a precast concrete panel with a diameter of 13 mm) were modeled using beam elements (T3D2), and 50-kg rails were modeled using solid elements (C3D8R). By applying the load factors suggested in [10], factored service loads were applied, and a static analysis was conducted.

Considering the objectives of this research, that is, the examination of the safety profile of the mountain-ERS and the analysis of the behavior of the entire structure under factored loads, some structural elements constituting the railway track system were not modeled based on several assumptions. First, we did not explicitly model the shear anchors in the structural analysis model because we assumed that their performance was optimal. This signifies the assumption that the interactions between the constituent structural elements were fully integrated, and that the entire structural system operated as a fully composite section. The tie option in the Abaqus program was used to integrate the structural elements. Nodes on different surfaces were paired to connect the translational and rotational degrees of freedom between the adjacent nodes. This was applied to the surfaces between the 50-kg rail and precast concrete panel, between the concrete panel and bedding layer, and between the bedding layer and subgrade in the structural analysis model. Owing to the assumption that no slip occurred between the layers, the lateral boundary conditions of these components were not determined.

Rails do not affect the strength or stiffness of railway track systems. However, the rails were modeled to consider the localized stresses that may be generated by concentrated train loads on the precast concrete panel, which do not occur in real structures. Similar to conventional railway track systems, synthetic resins act as rail fasteners, maintaining rail spacings and reducing vibrations. Therefore, based on the assumption that synthetic resins did not affect the safety of the railway track system, they were not modeled. Instead, a lateral load (wind load) transmitted from the rails to the concrete panel that did not induce local stresses was directly applied to the concrete panel as a distributed load with the same area as the synthetic resin material.

The rack rail installed at the center of the track system, which engaged with the pinions of mountain trains to transfer loads from the slope to the concrete panel, was not modeled based on the assumption that its effect on the safety and behavior of the concrete panel was not significant. Instead, the starting or braking loads transmitted by the rack rail to the concrete panel were applied at the points where the rack rails were connected to the concrete panel.

For the reinforcing bars placed inside a concrete panel, the embedded option in the Abaqus program was used in the analysis model to simulate the embedded reinforcing bars. If the node of the embedded element is located within the element containing it, the deformation degrees of freedom are removed, and the node functions as an embedded node. Using the corresponding constraint option, the internal nodes (i.e., the nodes of the embedded elements) can be configured for inclusion in a concrete panel.

The ground was not directly modeled to construct a structural analysis model of the railway track system. Instead, it was simulated as a boundary condition consisting of a fixed-end support condition and an elastic spring support condition. Detailed explanations are provided in Section 2.5.

2.3. Material Properties

2.3.1. Concrete

The material strength and modulus of elasticity of concrete can be obtained by directly testing a specimen or referring to the formulas presented in previous studies and design standards. For constituting the material properties of the concrete materials, that is, the precast concrete panel, mortar filler layer (bedding layer), and subgrade layer, a formula (obtained from the literature) [11,12] was used to calculate the values, including the modulus of elasticity according to the compressive strength, stress–strain relationship, and material properties. To calculate the elastic modulus of concrete materials, Equation (1) (presented in EN1992-1-1 [12]) was applied:

$$E_c=\mathrm{22,000}{\left(\frac{f_c^{″}}{10}\right)}^{0.3},$$

(1)

where $E_c$ is the elastic modulus of concrete, and $f_c^{″}$ is the maximum compressive stress at corresponding compressive strain.

The compressive strengths of the concrete and bedding layers were 45 MPa and 24 MPa, respectively. The Poisson’s ratio was assumed to be 0.18. The elastic modulus of each concrete component corresponding to its compressive strength is listed in

Table 2. Elastic modulus of concrete components by their corresponding compressive strengths.

Concrete Component	Compressive Strength	Elastic Modulus	Poisson’s Ratio
precast concrete panel	45 MPa	35,684 GPa	0.18
bedding layer	45 MPa	35,684 GPa	0.18
subgrade layer	24 MPa	30,160 GPa	0.18

.

The concrete damaged plasticity model was applied to the plastic behavior of concrete materials in the Abaqus program. The parameters for the model were assumed as reference values [11]: 36° for the dilation angle (ψ), 0.01 for eccentricity (ε), and 1.16 for the effective stress invariants value, which was calculated as the ratio $f_{b0}$/$f_{c0}$. The ratio of the second stress invariant on the tensile meridian ($K_c$) was assumed to be 0.667, and the viscosity parameter (μ) used for the viscoplastic regularization of the concrete constitutive equations was assumed to be equal to 0.0001.

For the compressive behavior of the concrete material, the stress–strain model presented by Hognestad [13] was applied. The stress–strain relationship for compression is shown in Figure 5.

For the ascending branch ($0\le {\epsilon }_c\le {\epsilon }_o$):

$$f_c=f_c^{″}\left[2\frac{\epsilon }{{\epsilon }_o}-{\left(\frac{\epsilon }{{\epsilon }_o}\right)}^2\right];$$

(2)

For the descending branch (${\epsilon }_o\le {\epsilon }_c<{\epsilon }_u$):

$$f_c=\left(\frac{-0.15f_c^{″}}{{\epsilon }_c-{\epsilon }_o}\right)\left(\epsilon -{\epsilon }_o\right)+f_c^{″},$$

(3)

where $f_c^{″}$ is the maximum compressive stress at compressive strain (${\epsilon }_o=2f_c^{″}/E_c$), $f_c$ is the compressive stress at strain (${\epsilon }_c$), and ${\epsilon }_u$ is the ultimate compressive strain, which can be set to 0.0038 with $0.85f_c^{″}$.

Concrete is a brittle material that is vulnerable to tension; therefore, its tensile behavior is determined by evaluating its tensile strength. When concrete is subjected to a tensile force, it exhibits a linear behavior up to the peak strain (${\epsilon }_{cr}$) at which cracks occur, and a nonlinear response thereafter. The stress tends to decrease as the strain increases. In the structural analysis model used in this research, the stress–strain model used for the tensile behavior of concrete was presented by Wang and Hsu (2001) [14]. An equation (presented in the CEB-FIP model code) was used for the peak tensile stress of the concrete [15] (Equation (6)). A linear relationship exists until the strain reaches the peak at which the crack occurs, and the part that exceeds the strain is presented as an exponential function. The stress–strain model for the tensile behavior is shown in Figure 6.

$$f_t=E_c{\epsilon }_t(0\le {\epsilon }_t\le {\epsilon }_{cr})$$

(4)

$$f_t=f_{cr}{\left(\frac{{\epsilon }_{cr}}{{\epsilon }_t}\right)}^{0.4}$$

(5)

$$f_{cr}=1.40\ast {\left(\frac{f_{ck}}{10\mathrm{MPa}}\right)}^{\left(2/3\right)}$$

(6)

where ${\epsilon }_{cr}$ is the tensile strain at which the tensile stress is equal to the peak stress ($f_{cr}$) value, $f_t$ is the tensile stress, ${\epsilon }_o$ is the compressive strain (${\epsilon }_o=2f_c^{″}/E_c$), $f_c$ is the compressive stress at strain (${\epsilon }_c$), and ${\epsilon }_u$ is the ultimate compressive strain, which can be set to 0.0038 with $0.85f_c^{″}$.

In this study, both linear and nonlinear material analyses were conducted. In the linear analysis, the elastic modulus of the concrete material corresponding to the compressive strength and Poisson’s ratio was applied for structural analyses.

2.3.2. Steel

The 50-kg rails and reinforcing bars inside the concrete panel were made of steel with yield stress of 800 Mpa and 400 Mpa, respectively. The elastic modulus of both components was 210,000 GPa, and the Poisson’s ratio was assumed to be 0.3. For each steel component, bilinear stress–strain models were applied, as shown in Equations (7) and (8) and Figure 7.

$$f_s=E_s{\epsilon }_s({\epsilon }_s\le {\epsilon }_y)$$

(7)

$$f_s=f_y({\epsilon }_s\le {\epsilon }_y)$$

(8)

For the linear analysis, the elastic moduli of the steel materials and their Poisson’s ratios were used. For the nonlinear analysis, including the material properties applied in the linear analysis, a bilinear material model was utilized, along with their yield strengths.

2.4. Load Conditions

Among the loads considered in the railway track design, those corresponding to the mountain-ERS track were selected and applied to the structural analysis model, as listed in

Table 3. Loads and corresponding factors considered in mountain-ERS analysis (Deutsches Institut für Normung (DIN) Technical Report-Actions on Bridges, 2009) [12].

Loads			Partial Safety Factor		Combination Coefficient
			ULS 1	SLS 2
dead load (self-wights)			1.35	1.00	1.00
live load (train loads)	applied at rails		1.45	1.00	1.00
starting/braking load	applied at rack rails		1.50	1.00	1.00
load owing to gradient			1.35	1.00	1.00
snake load (hunting oscillation)	applied at rails		1.50	1.00	1.00
wind load	applied at rails	applied with train loads simultaneously	1.50	1.00	1.00
loads owing to drying shrinkage			1.00	1.00	1.00

¹ Ultimate limit state. ² Service limit state.

. For the load factors corresponding to each load, the coefficients presented in the DIN Technical Report 101 (Actions on Bridges, 2009) [12] were applied. The loads, including the dead load, which is the self-weight of the components; the live load (train loads); the starting/braking load of the train; the loads owing to the gradient; and the wind load were considered. The snake load owing to train hunting was not considered because the mountain train moved at a low speed. In addition, the load arising from drying shrinkage was not considered because a precast concrete panel was utilized in the mountain-ERS track.

The loads listed in Table 3 were applied to the structural analysis model, as shown in Figure 8. They were applied in the vertical (y-), transverse (x-), and longitudinal (z-) directions.

For vertical loads, the analysis included both dead and live loads. The dead load included the weight of various components, including the 50-kg rails, precast concrete panels, synthetic resin materials, a mortar filler layer (bedding layer), and a subgrade layer. In this structural analysis model, the weights of these components were factored into the gravity loads based on their respective densities. To account for the 180‰ slope, the gravity loads were divided into two directions using this angle as a reference. As shown in Figure 9, the live train load was 90 kN. Similar to gravity loads, a live load was applied as a force component based on the gradient.

The wind load (transverse load) was calculated using the train dimensions. To calculate the wind load ($q_w$), the following equation presented in KR C 08020 [16] was used to calculate the wind load ($q_w$):

$$q_w=1.50·H,$$

(9)

where H denotes the height of the train, as shown in Figure 10. As discussed in Section 2.2, the transverse loads applied to the 50-kg rails were assumed to be distributed across the sections of the precast concrete panel.

In the case of longitudinal loads, the loads owing to the slope and starting/braking loads were considered, as shown in Figure 11. The longitudinal load ($P_G$) was calculated using Equation (10), as follows:

$$P_G=\left(P_t+P_D\right)\times \frac{S}{1000},$$

(10)

where $P_G$ denotes the longitudinal load, $P_t$ is the train load, $P_D$ is the dead load, and S is the slope. As previously mentioned, the longitudinal loads were divided into force components. For the starting/braking loads, the load occurring at one bogie of the train was calculated based on the length of the precast concrete panel and its system. The maximum starting and braking loads of the train were estimated by simulating the driving propulsion and brake performance of the train under development, as listed in

Table 4. Starting/braking loads of the mountain train under development for a gradient of 180‰.

Load	Calculated Load Value	Applied Load Value
starting load (180‰)	44.899 kN	89.798 kN
braking load (180‰)	38.150 kN

. In this structural analysis model, the starting/braking loads were selected conservatively based on the failures of the rack-rail pinion during operation. In the structural analysis model used in this study, the starting/braking load was applied as a distributed point load at intervals.

2.5. Boundary Conditions

The safety and structural behavior of railway track systems can be influenced by the ground conditions. Accordingly, the interaction between the railway track system and ground is crucial. In the structural analysis model, the presence of ground beneath the subgrade layer was assumed in two cases, and the differences between them were analyzed, as shown in Figure 12. First, it was assumed that the ground was sufficiently compacted during construction. Based on this assumption, the boundary condition of the bottom section of the subgrade layer was modeled with a fixed-end support ($U_x=U_y=U_z={UR}_x={UR}_y={UR}_z=0$). In the second case, considering the actual ground conditions at the site where the mountain-ERS was installed, an elastic spring support condition was assumed ($U_x=U_z={UR}_x={UR}_y={UR}_z=0$). In the study, the spring coefficient of the ground was considered and applied to the structural analysis model ($k_s=0.029$) [17].

3. Parametric Study for the Establishment of the Track Boundary Conditions of the Reference Model of the Mountain-ERS

To examine the safety of the mountain-ERS track structure subjected to service loads, the maximum principal tensile stresses of the concrete elements were analyzed by comparing the tensile strength values of the concrete materials based on their brittleness. The von Mises stress values were examined for reinforcing bars inside a precast concrete panel. The major points in the concrete elements were investigated by analyzing the structural behavior of the railway system, as shown in Figure 13. The stresses in each axis of the concrete components were defined as shown in this figure. S11, S22, and S33 denote the normal stresses perpendicular to the x, y, and z axes, respectively.

The vertical loads transmitted by the rails constitute the major loads of the mountain-ERS; therefore, points were chosen from a section where vertical loads were applied to the 50-kg rails. For the precast concrete panel, Point A (Point D) was in direct contact with the rails, Point B (Point E) was where the reinforcing bars were embedded, and Point C (Point F) was the lowest point, all of which were selected as review points. In the bedding layer, Points G (H) (C (F)) were investigated. In the case of the subgrade layer, the lowest points of the section were chosen as Points I and J. By investigating the vertical stress values of these points, the behavior of the structures was analyzed in each direction (S11, x-direction; S22, y-direction; S33, z-direction).

3.1. Mesh Convergence Analysis

Mesh convergence analysis was conducted to increase the reliability of the results of the structural analysis model for mountain-ERS. Considering the geometries of the structural analysis model sections, a mesh convergence analysis was conducted on the following numbers of elements: 204,384 (element size = 25 mm), 137,078 (element size = 30 mm), 92,124 (element size = 34 mm), 61,014 (element size = 40 mm), 42,644 (element size = 45 mm), and 37,384 (element size = 25 mm). Point B was selected as the point to be evaluated for the mesh convergence analysis, where the longitudinal reinforcing bars were arranged inside the concrete panel. The results varied considerably, depending on the size of the elements and whether reinforcing bars were included. The value of S22 was the largest among the other stresses; therefore, the values at Point B were evaluated using the analyzed cases. As shown in Figure 14, as the number of elements increased, S22 tended to converge to a specific value. To obtain reliable results with an efficient structural analysis, a structural analysis model with 92,124 elements (element size = 34 mm) was chosen as the reference model.

3.2. Examination of the Nonlinear Effects on the Structural Behavior of the Mountain-ERS Track

To simulate the precise behavior of the brittle characteristics of concrete materials, the material nonlinearities of the concrete components were applied in the structural analysis model. In addition, because the magnitudes of the stresses estimated in the mesh convergence analysis were within the linear elastic limit of concrete, a linear structural analysis model was constructed in which all the structural elements were assumed to be elastic. The results of the nonlinear and linear structural analysis models are shown in Figure 15 and listed in

Table 5. Differences of normal stresses of nonlinear and linear analysis.

Points		Nonlinear and Linear Analyses
		S11			S22			S33
		Nonlinear Analysis	Linear Analysis	Error	Nonlinear Analysis	Linear Analysis	Error	Nonlinear Analysis	Linear Analysis	Error
left rail	A	4.77 × 10−1	4.77 × 10−1	0.00%	−3.31038	−3.31038	0.00%	2.76 × 10−1	2.76 × 10−1	0.00%
	B	1.46 × 10−1	1.46 × 10−1	0.00%	−1.14192	−1.14192	0.00%	−4.34 × 10−3	−4.34 × 10−3	0.08%
	C	1.18 × 10−1	1.18 × 10−1	0.00%	−6.96 × 10−1	−6.96 × 10−1	0.00%	−1.44 × 10−2	−1.44 × 10−2	0.02%
	F	1.04 × 10−1	1.04 × 10−1	0.00%	−4.90 × 10−1	−4.90 × 10−1	0.00%	2.72 × 10−3	2.71 × 10−3	−0.11%
	H	−6.38 × 10−4	−6.38 × 10−4	0.00%	−7.19 × 10−2	−7.19 × 10−2	0.00%	−1.24 × 10−2	−1.24 × 10−2	0.00%
right rail	D	4.31 × 10−1	4.31 × 10−1	0.00%	−3.34144	−3.34144	0.00%	2.66 × 10−1	2.66 × 10−1	0.00%
	E	1.47 × 10−1	1.47 × 10−1	0.00%	−1.13637	−1.13637	0.00%	−3.13 × 10−3	−3.12 × 10−3	−0.09%
	F	1.28 × 10−1	1.28 × 10−1	0.00%	−7.05 × 10−1	−7.05 × 10−1	0.00%	−1.41 × 10−2	−1.41 × 10−2	−0.02%
	G	1.06 × 10−1	1.06 × 10−1	0.00%	−4.85 × 10−1	−4.85 × 10−1	0.00%	5.59 × 10−3	5.60 × 10−3	−0.04%
	I	−3.61 × 10−3	−3.61 × 10−3	0.00%	−7.60 × 10−2	−7.60 × 10−2	0.00%	−1.25 × 10−2	−1.25 × 10−2	0.00%

Unit: MPa.

. The stress distribution and tendency of the major points were similar in both structural analysis models. Upon comparison of the results of the two analysis models, it was found that the differences in the values at the points selected for the analysis were sufficiently small. This was because the stresses inside the precast concrete panel, bedding layer, and subgrade layer did not exceed the linear elastic limit of each material. Based on these results, a structural analysis was performed using linear elastic material models.

3.3. Influences of Different Boundary Conditions on the Analyzed Results

As mentioned in Section 2.5, the safety profile and structural behavioral response of the railway track system can be influenced by ground conditions; therefore, two boundary condition cases, the fixed-end and spring support, were set for the structural analysis model. The results for both the boundary conditions are shown in Figure 16. In both cases, there were no significant differences in the stress trends and their structural deformations were similar. With respect to S22, the effects of the factored load on the occurrence of S22 and distribution along the structural elements were similar in both cases. However, the tendencies of S11 and S33 exhibited minor differences in both cases. These differences occurred because of the different boundary conditions. In the fixed-end boundary condition model, the bottom section of the subgrade layer could not move along the y-direction or be deformed. However, in the spring-support model, the structural element was deformable in the vertical direction (y-axis) relative to the ground coefficient.

The occurrence trends of S11 and S33 differed because the deformations in the rail differed in the longitudinal and transverse directions when the train load was applied. Although there was a slight difference in the trends of S11 and S33 in both cases, the difference was not statistically significant. In this structural analysis model, a reference model was constructed using an elastic spring support condition that reflected actual ground conditions.

4. Analytical Investigation of the Behavioral Characteristics of the Mountain-ERS Track System

4.1. Structural Safety Evaluation of the Mountain-ERS under Factored Service Loading Conditions

As discussed previously, the safety of the mountain-ERS track system was also evaluated. The maximum principal stress of each concrete element and the von Mises stresses of the reinforcing bars were investigated, as shown in Figure 17 and Figure 18, respectively. The estimates of the maximum principal tensile stress of each concrete component were obtained as follows: 3.910 MPa in the middle section of the concrete panel at the position where the rack rail was installed, 0.404 MPa in the bedding layer, and 0.425 MPa in the subgrade layer.

The maximum principal stress of the concrete elements did not exceed the tensile strength of the concrete, except at the surface, where discrete longitudinal forces were applied to the precast concrete panel. This was because of the postulated assumption for the conservative safety assessment of mountain-ERS, which considered the starting and braking loads of the mountain train. In addition, the differences between the values of the maximum principal tensile stress (3.910 MPa) and tensile strength of the concrete panel (3.81 MPa) were negligible. In the case of the starting and braking loads of the mountain train, an arbitrarily applied value of the prototype train was applied in this research; therefore, it is necessary to evaluate safety and calculate and apply the exact values based on additional future research.

For the reinforcing bars, a von Mises stress of 36.93 MPa was generated at the position where the longitudinal loads on the rack rail were installed, which was lower than the yield strength of the reinforcing bars (400 MPa).

4.2. Feasibility Study of High-Strength Concrete Material for Enhancing the Structural Safety of the Mountain-ERS Track

In the ERS track, both train and wind loads were transmitted from the rails to the precast concrete panels. Longitudinal loads, such as braking forces, were transmitted from a rack rail to a precast concrete panel. Precast concrete panels must possess sufficient strength to withstand such loads. The bedding and subgrade layers transferred the loads transmitted from the precast concrete panels to the underlying ground.

To analyze the effect of the strength of the precast concrete panel and the behavior of the mountain-ERS track under these conditions, parametric case studies were conducted by changing the material properties of the precast concrete panel. Linear analyses were conducted, and the elastic modulus of the concrete was adjusted with respect to its compressive strength. The compressive strengths of the precast concrete panel were assumed to be 45 MPa (35,684 GPa, reference model), 60 MPa (38,776 GPa), 75 MPa (41,440 GPa), and 100 MPa (45,241 GPa), and the corresponding elastic moduli were considered based on Equation (1) ([12]) (the values in parentheses indicate the modulus of elasticity of concrete). The results are presented in Figure 19.

When a high-strength concrete material was applied to the precast concrete panels, the tendency of S22 to occur along the sections was similar. However, for S11 and S33, the stress tended to increase at the bottom of the panel when high-strength concrete was used. These tendencies were attributed to the difference in the elastic modulus of the precast concrete panel from those of the reinforcing bars, bedding layer, and subgrade layer.

For the same reasons, as the elastic modulus of the precast concrete panel increased, the maximum principal tensile stress within the concrete components increased, whereas the maximum principal tensile stress in the bedding and subgrade layers decreased, as indicated in

Table 6. Maximum principal tensile stresses of concrete elements and von Mises stress of reinforcing bars inside the concrete panel (parametric study cases: the compressive strength (elastic modulus) of the precast concrete panel).

Compressive Strength of the Precast Concrete Panel (Elastic Modulus) (Tensile Strength)	45 MPa (35,685 GPa) (3.82 MPa) (Reference)	60 MPa (38,776 GPa) (4.62 MPa)	75 MPa (41,440 GPa) (5.36 MPa)	100 MPa (45,241 GPa) (6.50 MPa)
concrete panel	3.91	3.99	4.04	4.12
bedding layer	4.04 × 10−1	3.95 × 10−1	3.87 × 10−1	3.77 × 10−1
subgrade layer	4.25 × 10−1	4.15 × 10−1	4.07 × 10−1	3.96 × 10−1
reinforcing bars	3.69 × 101	3.54 × 101	3.41 × 10−1	3.25 × 101

Unit: MPa.

. The occurrence patterns of the maximum principal tensile stress in the concrete components and the location of the maximum principal tensile stress are depicted in Figure 20, whereas the von Mises contours and the location of the maximum tensile stress in the reinforcing bars are shown in Figure 21. Similar trends in the stress contours of the concrete components and von Mises contours of the reinforcing bars were observed in each case. In all the cases, the maximum principal tensile stress did not exceed the tensile strength of the concrete components.

4.3. Effects of the Gradient Variation on the Stress of the Mountain-ERS Track

By varying the gradient conditions of the mountain-ERS, safety assessments and behavioral analyses were conducted, including gradients of 0‰, 50‰, 100‰, 150‰, 180‰ (reference model), and 200‰. As described in Section 2.4, the gradient variations resulted in loads applied in the form of divided loads at these angles. Consequently, the self-weight and train loads were adjusted to consider the variations in the gradients of the structural system.

The stress inside this section is shown in Figure 22. In the case of S22, slight differences in the values were observed at the position where the precast concrete panel contacted the 50-kg rails as the gradient of the mountain-ERS changed; the stresses, on the other hand, in the bedding and subgrade layers exhibited similar trends. For S11, a trend similar to that of S22 was observed; the tensile stresses increased slightly as the gradient of the mountain-ERS decreased, particularly when the precast concrete panel contacted the rail.

For S33, significant changes were observed in the precast concrete panels. This is because an increase in the gradient triggered a significant increase in the longitudinal forces acting on the mountain-ERS. Specifically, at a gradient of 0‰, the largest difference from the reference model (180‰) was observed as the divided force in the transverse direction became zero.

In each case, the magnitude of the maximum principal tensile stress in the concrete components did not exceed the tensile strength of the concrete, as listed in

Table 7. Maximum principal tensile stresses of concrete elements and von Mises stresses of reinforcing bars inside the concrete panel (parametric study cases: variation of gradients).

Gradient (‰)	0	50	100	150	180	200
concrete panel	3.90	3.91	3.91	3.91	3.91	3.91
bedding layer	4.02 × 10−1	4.03 × 10−1	4.04 × 10−1	4.04 × 10−1	4.04 × 10−1	4.04 × 10−1
subgrade layer	4.25 × 10−1	4.25 × 10−1	4.25 × 10−1	4.25 × 10−1	4.25 × 10−1	4.25 × 10−1
reinforcing bars	3.69 × 101	3.69 × 101	3.69 × 101	3.69 × 101	3.69 × 101	3.69 × 101

Unit: MPa.

. The contours of the principal stress of concrete and the location of the maximum principal tensile stress with variations in the gradient of the mountain-ERS are shown in Figure 23. The von Mises stress contours and location of the maximum value of the reinforcing bars are shown in Figure 24. The occurrence of stress and the location of the maximum principal tensile stress were similar, and their values were also similar to the variation in the gradients.

4.4. Effects of Thickness of the Subgrade Layer on the Stress of the Mountain-ERS Track

Based on the results presented above, it was concluded that the precast concrete panels of the mountain-ERS exhibited sufficient robustness against factored service loads. Based on this consideration, safety assessment and behavioral analysis of mountain-ERS were conducted by reducing the thickness of the subgrade layer. A parametric study was conducted by reducing the thickness of the subgrade of the reference model (total thickness = 200 mm) at 25-mm intervals. This study accounted for thickness reductions of −25 mm (subgrade thickness:175 mm), −50 mm (subgrade thickness:150 mm), −75 mm (subgrade thickness:125 mm), and −100 mm (subgrade thickness:100 mm). The stress distributions in the track sections are shown in Figure 25.

As the thickness of the subgrade layer decreased, the distribution of S22 generally exhibited a consistent trend in each case study. The vertical loads applied to the rails were the same in the parametric cases; the distributions of S22 exhibited similar trends.

For S11, the tensile stresses in the precast concrete panel, bedding layer, and subgrade tended to decrease with the increasing thickness of the subgrade layer. Similarly, for S33, the tensile stress decreased, whereas the compressive stress increased. This is attributable to the conditions applied to the structural analysis model. A tie option that enabled each concrete component to be fixed to the connected surface was used to prevent slippage between them. Additionally, owing to the boundary conditions applied at the bottom section of the subgrade, localized behavior occurred in these sections.

The principal stress contours and the location of the maximum principal stress of the concrete elements are shown in Figure 26. The von Mises stress contour and location of the maximum stress value of the reinforcing bars are shown in Figure 27. The stress distribution and locations of the maximum principal tensile stress within the concrete components and the locations of the reinforcing bars were similar in all studied cases; the values are listed in

Table 8. Maximum principal tensile stresses of concrete elements and von Mises stresses of reinforcing bars inside the concrete panel (parametric study cases wherein the thickness of the subgrade layer decreased at set intervals).

Thickness Decrease in the Subgrade Layer	0 mm (Reference)	−25 mm	−50 mm	−75 mm	−100 mm
concrete panel	3.91	3.91	3.91	3.92	3.92
bedding layer	4.04 × 10−1	4.16 × 10−1	4.34 × 10−1	4.39 × 10−1	4.44 × 10−1
subgrade layer	4.25 × 10−1	4.26 × 10−1	4.33 × 10−1	4.79 × 10−1	5.03 × 10−1
reinforcing bars	3.69 × 101	3.70 × 101	3.70 × 101	3.71 × 101	3.71 × 101

Unit: MPa.

. The results of these analyses indicated that even when the thickness of the subgrade of the mountain-ERS track was reduced to half of its original value, the maximum tensile stress in the concrete members did not exceed the tensile strength of the concrete materials.

5. Conclusions

In this study, a three-dimensional structural analysis model of a mountain-ERS track with a steep gradient of 180‰ was established. The ground coefficient was applied to simulate the ground conditions at the site where the mountain railway was constructed. By applying service loads to the factors presented in the DIN Technical Report 101 [10], the safety of mountain-ERS was evaluated, and their structural behaviors were analyzed.

• The safety assessment revealed that the maximum principal tensile stress in the precast concrete panel, bedding layer, and subgrade layer of the mountain-ERS did not exceed the tensile strength of the concrete under the applied factored loading conditions. The stress values inside the concrete elements did not exceed their linear elastic limits. The mountain-ERS did not exhibit overall flexural deformation, and localized behavior was observed in the sections where the train load was applied to the rails.
• With respect to the application of high-strength concrete materials for precast concrete panels, the maximum principal tensile stresses of the concrete elements increased as the elasticity modulus of the concrete increased, whereas those of the bedding and subgrade layers decreased. Applying a high-strength concrete material to the reinforcing bars inside the precast concrete panel reduced the von Mises stresses. The variations in S11 and S33 in the lower part of the concrete panel (close to the bedding layer) were attributed to the differences in the elasticity modulus. Under the load conditions considered in this structural analysis, the application of high-strength concrete material to the precast concrete panel resulted in a slight increase in the internal stresses within the concrete components, although this effect was not prominent.
• The trends in the occurrence of S22 (vertical direction) and S11 (transverse direction) were similar when the gradient was varied. However, they induced a considerable variation in S33 (in the longitudinal direction). As the gradient of the mountain-ERS changed, the magnitude of the longitudinal component of the train load transmitted from the rail also changed considerably, indicating a difference in S33 in the uppermost part (Point A) of the precast concrete panel. The stress trends in the bedding and subgrade layers were similar. For each variable, the magnitude of the maximum principal tensile stress inside the concrete member did not exceed its tensile strength.
• No significant differences were observed in the occurrence of S22 (in the vertical direction) when the thickness of the subgrade layer of the mountain-ERS track decreased. However, noticeable changes were observed at sites S11 and S33 in the lowest part of the subgrade layer. Even with a reduced subgrade layer thickness (compared with the initial thickness), the maximum principal tensile stresses within each concrete member did not exceed the tensile strength of the concrete material when service loads with factors were applied in the structural analysis model of this study. Based on the above results, it was concluded that the precast concrete panels of the mountain-ERS exhibited sufficient robustness against factored service loads.
• The analysis methods applied in the structural analysis model and the results derived in this research can be utilized as basic research data for the analysis of the behavior of mountain-ERS and the optimization of sections of mountain-ERS tracks.