**6. Ballast layer modeling**

Three-dimensional shape measurement was performed to ascertain three-dimensional vertex coordinates of more than 4000 ballast gravel pieces. Based on the measured coordinates, each shape was expressed using a polyhedral rigid-body discrete element model and was converted into aggregate, with the same size and the same shape, of the tetrahedron secondary elastic finite elements. Details of the measurements were presented in an earlier report [28]. **Figure 10** presents some exemplary images of a ballast gravel piece and its numerical discrete element and finite element models. Regarding the finite element models, a sufficiently fine

**Figure 10.** Exemplary pictures of existing ballasts and the digitized models. (a) Ballast, (b) DEM model, (c) FEM model.

mesh size of 1 cm was adopted to support the precise representation of natural frequencies of individual ballast gravel up to several 10 kHz.

position in 1.1 ms. Results show that only about 1 ms is necessary for the ballast layer compression and restoration. Displacement responses of the sleeper and load during that time

**Figure 9.** Vertical displacement of sleepers immediately after impact loading. (a) Sleeper displacement, (b) close-up of

Following the compression and restoration motions of the ballast layer, sleeper-jumping occurs. The average jumping speed is approximately 1.71 times as high as the average of the compression speed. Their initial speed can be approximately 12.1 times faster than the average compression speed. The jumping height of the sleeper reaches 5.1 mm in 24.4 ms. The sleeper returns to the initial position within 46.7 ms after loading. The jumping behaviour during unloading includes no high-frequency vibration component. Most of the vibration comprises low-frequency components. The author posits that the cause of sleeper-jumping is

**Figure 9**, as described, presents the vertical displacement of the sleeper. Presumably, similar behaviour occurs in the upper part of the ballast layer immediately under the sleepers, which means that the motion under impact loading is extremely slight. The compression and restoration behaviours are high-frequency responses that last for a very short time: about 1 ms. Therefore, high-frequency vibration components are dominant in ballast responses under loading. However, ballast motions during unloading are induced mainly by low-frequency

Three-dimensional shape measurement was performed to ascertain three-dimensional vertex coordinates of more than 4000 ballast gravel pieces. Based on the measured coordinates, each shape was expressed using a polyhedral rigid-body discrete element model and was converted into aggregate, with the same size and the same shape, of the tetrahedron secondary elastic finite elements. Details of the measurements were presented in an earlier report [28]. **Figure 10** presents some exemplary images of a ballast gravel piece and its numerical discrete element and finite element models. Regarding the finite element models, a sufficiently fine

include few vibration components at low frequencies.

displacement.

112 New Trends in Structural Engineering

the abrupt release of the strain energy stored in the ballast.

**6. Ballast layer modeling**

vibration components that cause large displacement and which last longer.

Regarding the physical properties of the ballast, density *ρ* = 2700 (kg/m3 ), Young's modulus *E* = 30 (GPa), Poisson's ratio *ν* = 0.2 and structural damping parameter *η* = 0.01 are adopted. The ballast gravel density was the laboratory experimental value obtained from specific gravity tests. The Young's moduli and Poisson's ratio were referred or derived from previous reports of the literature. Regarding the structural damping coefficient, the author adopted general values of a concrete structure.

**Figure 11** presents a procedure for the creation of the ballast aggregate using both discrete element and finite element modeling. First, about 100 pieces of the ballast polyhedron discrete element models with different shapes and sizes were placed randomly in the air above rectangular box frames of 20 cm width and length. The gravel was then dropped freely with gravity and was compressed vertically with a loading plate using discrete element software. Next, all individual polyhedron discrete element models were converted into assemblages consisting of the small finite tetrahedron second-order elastic solid elements and split into 1 cm meshes with geometry and contact point information maintained. Each polyhedron discrete element model was divided into approximately 1000 tetrahedron finite elements. The finished rectangular block model has 20 cm width and length, with 17 cm height, and has more than 90,000 tetrahedron finite elements.

**Figure 11.** Compaction and modeling of the ballast aggregate with MPCs.

For this procedure, the contact points were mutually connected through multipoint constraints (MPCs), which connected the nodes of elements in three axial directions related to the contact pair of blocks at each contact point. The provided contact-connectivity exhibited no expansion or contraction because it was several tens of thousands of times harder than that of ballast pieces. Accordingly, the spring functions around the contact points were represented by the elastic deformation of the ballast angularities, which were composed of the assemblages of several tetrahedron finite elements adjacent to the contact points.

**Figures 12** and **13** present a finite element model of a type 3 PC monoblock sleeper developed by the authors [24] and consisting of 51,146 nodes and 51,944 solid elements. The sleeper model can make a precise representation of all natural frequencies described above, with their response values up to 1 kHz within 5% deviation. Its physical weight is 161.40 kg. Its volume is 0.0677 m3 .

A large-scale finite element model was constructed with multipoint constraints (MPCs) by assembling 48 units of the rectangular unit block models of the ballast described above and

the type 3 PC sleeper model. **Figure 14** presents finished analysis model and its domain segmentation for large-scale parallel computing by the finite element programming code of FrontISTR. The model consists of 7.05 million nodes and 4.15 million second-order solid elements. The degrees of freedom of the model exceed 21 million. By adopting the finite element analysis using the precise model of the ballast layer having a complicated structure, it is possible to reproduce phenomena such as stress concentration and wave propagation within the ballast layer rapidly, easily and exactly. The entire model is divided into 24 sub-domains to introduce the direct solver MUMPS corresponding to the parallel computing of a distributed memory type. By applying the finite element normal-mode analysis by FrontISTR, a set of normal modes related to the ballast aggregate and sleeper system is obtained. Specifications

**)**

Vertical Natural Vibration Modes of Ballasted Railway Track

http://dx.doi.org/10.5772/intechopen.79738

115

**Figure 14.** Assembly of ballasted track model and domain segmentation for large-scale parallel computing.

**Item Young's modulus (GPa) Poisson's ratio Density (kg/m3**

Ballast 30 0.200 2700 Concrete 45 0.167 2350 Steel 210 0.290 7820

related to the various parameters of each model are presented in **Table 1**.

the top surface of the previously described ballasted track model.

**vibration mode of the ballasted track**

120 km/h [26, 27].

**Table 1.** Calculation parameters.

**7. Large-scale finite element transient response analysis of the elastic** 

The time history response waveforms were calculated numerically by inputting the measured loading waveforms to the top surface of a sleeper model when a passenger train moved over

**Figure 15** displays the actual waveforms of vertical loading (measured in cross-sectional area of 14 cm width and 18 cm length on the bottom surface of the rails) applied by the first axles of a lead coach bogie when the passenger train moved through the test section at about

**Figure 12.** Model of a type 3 PC sleeper.

**Figure 13.** Model of a type 3 PC sleeper. (a) Side view, (b) top view, (c) end face and reinforce bars.

**Figure 14.** Assembly of ballasted track model and domain segmentation for large-scale parallel computing.


**Table 1.** Calculation parameters.

**Figure 13.** Model of a type 3 PC sleeper. (a) Side view, (b) top view, (c) end face and reinforce bars.

For this procedure, the contact points were mutually connected through multipoint constraints (MPCs), which connected the nodes of elements in three axial directions related to the contact pair of blocks at each contact point. The provided contact-connectivity exhibited no expansion or contraction because it was several tens of thousands of times harder than that of ballast pieces. Accordingly, the spring functions around the contact points were represented by the elastic deformation of the ballast angularities, which were composed of the assem-

**Figures 12** and **13** present a finite element model of a type 3 PC monoblock sleeper developed by the authors [24] and consisting of 51,146 nodes and 51,944 solid elements. The sleeper model can make a precise representation of all natural frequencies described above, with their response values up to 1 kHz within 5% deviation. Its physical weight is 161.40 kg. Its volume

A large-scale finite element model was constructed with multipoint constraints (MPCs) by assembling 48 units of the rectangular unit block models of the ballast described above and

blages of several tetrahedron finite elements adjacent to the contact points.

is 0.0677 m3

.

114 New Trends in Structural Engineering

**Figure 12.** Model of a type 3 PC sleeper.

the type 3 PC sleeper model. **Figure 14** presents finished analysis model and its domain segmentation for large-scale parallel computing by the finite element programming code of FrontISTR. The model consists of 7.05 million nodes and 4.15 million second-order solid elements. The degrees of freedom of the model exceed 21 million. By adopting the finite element analysis using the precise model of the ballast layer having a complicated structure, it is possible to reproduce phenomena such as stress concentration and wave propagation within the ballast layer rapidly, easily and exactly. The entire model is divided into 24 sub-domains to introduce the direct solver MUMPS corresponding to the parallel computing of a distributed memory type. By applying the finite element normal-mode analysis by FrontISTR, a set of normal modes related to the ballast aggregate and sleeper system is obtained. Specifications related to the various parameters of each model are presented in **Table 1**.
