**7. Large-scale finite element transient response analysis of the elastic vibration mode of the ballasted track**

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 the top surface of the previously described ballasted track model.

**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 120 km/h [26, 27].

**Figure 15.** Measured rail seat loads.

The figure shows that the axles passed immediately above the sleeper centre at around 55 ms. Finite element analysis shows that the measured time history waveforms are uniformly input to all nodal points located within the bottom area of the rails on the top surface of the sleeper model. The calculation time interval is set at Δ*t* = 0.1 (ms). The total calculation steps are 800.

upward (compression) and downward (tension) motions at a frequency of approximately 300 Hz, according to the inputted loads. In this case, the maximum stresses of approximately 50 MPa after elapsed time of about 45–53 ms are observed near the ballast surface beneath the sleeper bottom. It is apparent that the rise in the peak response value becomes gentler as the measurement point becomes increasingly distant (i.e. deeper) from the loading point. A tendency exists for waveforms to become smoother along with the steep decrease in high-frequency vibration included in the wave. Although the investigation reported here involves elastic body analysis without the use of any constitutive equation, the diminishing trend of energy inside the ballast is reproduced closely by simulating the ballast aggregate structure in detail. Results show that the ballast aggregate structure featuring angular parts has the mechanism of energy attenuation. **Figure 18** portrays linear amplitude spectra of response of the von Mises stress on the crosssection inside the gravel angle acting at different depths below the loading point of the left rail, as obtained by conducting fast Fourier transformation of these time history response waveforms and smoothing them at a 20 Hz bandwidth. As the figure shows, these spectra curves identify the first-order elastic vibration resonance mode of the ballast layer at around

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337 Hz, where the whole ballast layer stretches vertically as an elastic body.

times the value of the measured natural frequency of the rigid-body mode.

ballast aggregate model.

**Figure 17.** Time history of on Mises stress.

To detect the exact frequency of the elastic vibration mode of the ballast layer, the author conducted a numerical experiment (simulation), which simulates the impulse loading experiment using a transient response analysis of the previously described large-scale sleeper-

**Figure 19** shows the amplitude spectra of response of the vertical displacement of the sleeper by application of a 0.1 ms square waveform of impulse loading. Impulse loading to the left and right rails totals 100 kN. Results of the numerical experiment of the impact loading indicate that the rigid-body natural vibration mode occurs at around 310 Hz. Results show that the analytical natural frequency of the elastic vibration mode of the ballast layer more or less coincides with the measured one described above and that the frequency corresponds to three

**Figure 16** portrays the distribution of the response nodal displacement of the ballast aggregate and the sleeper system at *t* = 55.0 (ms) when the load peaks appeared as the first axle of the lead coach bogie that passed above the sleeper centre. It is apparent that the dynamic displacement induced by a passing train on the ballast is not distributed uniformly throughout the ballast aggregate and the sleeper. Significant displacement is concentrated locally around the rail positions. Analytical results demonstrate that 30 μm maximum downward displacement occurred.

**Figure 17** depicts the time history waveforms of response of the von Mises stress on the crosssection inside the gravel angle acting on three contact points, each with some angularity, of the ballast gravel placed at different depths below the loading point of the left rail location. As the figure shows, the stresses in the gravel increase gradually, alternately repeating the minute

**Figure 16.** Distribution of nodal displacement (*t* = 55.0 ms).

**Figure 17.** Time history of on Mises stress.

The figure shows that the axles passed immediately above the sleeper centre at around 55 ms. Finite element analysis shows that the measured time history waveforms are uniformly input to all nodal points located within the bottom area of the rails on the top surface of the sleeper model. The calculation time interval is set at Δ*t* = 0.1 (ms). The total calculation steps are 800. **Figure 16** portrays the distribution of the response nodal displacement of the ballast aggregate and the sleeper system at *t* = 55.0 (ms) when the load peaks appeared as the first axle of the lead coach bogie that passed above the sleeper centre. It is apparent that the dynamic displacement induced by a passing train on the ballast is not distributed uniformly throughout the ballast aggregate and the sleeper. Significant displacement is concentrated locally around the rail positions. Analytical results demonstrate that 30 μm maximum downward displacement occurred. **Figure 17** depicts the time history waveforms of response of the von Mises stress on the crosssection inside the gravel angle acting on three contact points, each with some angularity, of the ballast gravel placed at different depths below the loading point of the left rail location. As the figure shows, the stresses in the gravel increase gradually, alternately repeating the minute

**Figure 15.** Measured rail seat loads.

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**Figure 16.** Distribution of nodal displacement (*t* = 55.0 ms).

upward (compression) and downward (tension) motions at a frequency of approximately 300 Hz, according to the inputted loads. In this case, the maximum stresses of approximately 50 MPa after elapsed time of about 45–53 ms are observed near the ballast surface beneath the sleeper bottom. It is apparent that the rise in the peak response value becomes gentler as the measurement point becomes increasingly distant (i.e. deeper) from the loading point. A tendency exists for waveforms to become smoother along with the steep decrease in high-frequency vibration included in the wave. Although the investigation reported here involves elastic body analysis without the use of any constitutive equation, the diminishing trend of energy inside the ballast is reproduced closely by simulating the ballast aggregate structure in detail. Results show that the ballast aggregate structure featuring angular parts has the mechanism of energy attenuation.

**Figure 18** portrays linear amplitude spectra of response of the von Mises stress on the crosssection inside the gravel angle acting at different depths below the loading point of the left rail, as obtained by conducting fast Fourier transformation of these time history response waveforms and smoothing them at a 20 Hz bandwidth. As the figure shows, these spectra curves identify the first-order elastic vibration resonance mode of the ballast layer at around 337 Hz, where the whole ballast layer stretches vertically as an elastic body.

To detect the exact frequency of the elastic vibration mode of the ballast layer, the author conducted a numerical experiment (simulation), which simulates the impulse loading experiment using a transient response analysis of the previously described large-scale sleeperballast aggregate model.

**Figure 19** shows the amplitude spectra of response of the vertical displacement of the sleeper by application of a 0.1 ms square waveform of impulse loading. Impulse loading to the left and right rails totals 100 kN. Results of the numerical experiment of the impact loading indicate that the rigid-body natural vibration mode occurs at around 310 Hz. Results show that the analytical natural frequency of the elastic vibration mode of the ballast layer more or less coincides with the measured one described above and that the frequency corresponds to three times the value of the measured natural frequency of the rigid-body mode.

With finite element analysis, the measured waveforms of the rail seat load were inputted to the top surface of the sleeper model using tensionless springs. **Figure 20** presents the nodal displacement distribution of the ballast aggregate and the sleeper system at *t* = 55.0 ms when load peaks appeared immediately after the first axle of the lead coach bogie passed above the

**Item Compression spring factor (GN/m) Tension spring factor (GN/m)**

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Ballast-ballast 30 0.0003 Sleeper-ballast 10 0.0001

**Table 2.** Calculation parameters related to contact points.

**Figure 20.** Distribution of nodal displacement (*t* = 55.0 ms).

**Figure 21.** Time history response of the von Mises stress.

**Figure 21** depicts time history waveforms of the response of the von Mises stress on the cross-section inside the angular part of the gravel under the left rail. As the figure shows,

sleeper centre.

**Figure 18.** Response spectrum of the von Mises stress.

**Figure 19.** Response spectrum of vertical displacement regarding impulse loading.

## **8. Large-scale finite element transient response analysis related to the rigid-body vibration mode of the ballasted track**

To examine the dominant rigid-body resonance motion around 100 Hz, this chapter presents a description of the large-scale finite element transient response analysis of the ballasted track using nonlinear contact springs (i.e. tensionless springs) in place of the MPCs. Results of the drop-weight tests described above suggest that the jumping motion of the ballasted track, the rigid-body bounce mode, will cause large displacement. Therefore, regarding the previously described large-scale finite element model of the ballasted track, information of the contact points between the ballast pieces and the sleeper nodes is modeled with nonlinear contact springs, that is, with tensionless contact. **Table 2** presents parameters of calculations related to contact points. In this analysis, the tension spring factors are set at 1/100,000 values of the compression spring factors.


**Table 2.** Calculation parameters related to contact points.

With finite element analysis, the measured waveforms of the rail seat load were inputted to the top surface of the sleeper model using tensionless springs. **Figure 20** presents the nodal displacement distribution of the ballast aggregate and the sleeper system at *t* = 55.0 ms when load peaks appeared immediately after the first axle of the lead coach bogie passed above the sleeper centre.

**Figure 21** depicts time history waveforms of the response of the von Mises stress on the cross-section inside the angular part of the gravel under the left rail. As the figure shows,

**Figure 20.** Distribution of nodal displacement (*t* = 55.0 ms).

**8. Large-scale finite element transient response analysis related to** 

To examine the dominant rigid-body resonance motion around 100 Hz, this chapter presents a description of the large-scale finite element transient response analysis of the ballasted track using nonlinear contact springs (i.e. tensionless springs) in place of the MPCs. Results of the drop-weight tests described above suggest that the jumping motion of the ballasted track, the rigid-body bounce mode, will cause large displacement. Therefore, regarding the previously described large-scale finite element model of the ballasted track, information of the contact points between the ballast pieces and the sleeper nodes is modeled with nonlinear contact springs, that is, with tensionless contact. **Table 2** presents parameters of calculations related to contact points. In this analysis, the tension spring factors are set at 1/100,000 values of the

**the rigid-body vibration mode of the ballasted track**

**Figure 19.** Response spectrum of vertical displacement regarding impulse loading.

**Figure 18.** Response spectrum of the von Mises stress.

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compression spring factors.

**Figure 21.** Time history response of the von Mises stress.

considerable stresses of approximately 57 MPa are obtained near the surface of the ballast layer under the sleeper bottom. In the figure, in general, the positive motion denotes compression and downward behaviour. Negative motion denotes tension and upward behaviour.

**Figure 22** displays the linear amplitude spectra of response of von Mises stress, on the crosssection inside the gravel angle acting at different depths below the loading point of the left rail, which are obtained by conducting fast Fourier transformation of these time history response waveforms and by smoothing them at 20 Hz bandwidth. As the figure shows, these spectral curves identify the first-order elastic vibration resonance mode of the ballast layer at around 337 Hz, where the entire ballast layer stretches vertically as an elastic body.

**Figure 23** presents the time history of response of the vertical displacement at the left edge of the sleeper immediately after the 0.1 ms square impulse loading waveforms of 100 kN. In this figure, the downward displacement shows the downward motion of the sleeper (i.e. ballast

> layer compression); the upward displacement represents the upward motion of sleeper (i.e. ballast layer extension). According to the figure, when an impact load is applied, the ballast layer instantaneously deforms elastically because of compression. It then returns to the preloading position. Consequently, it takes only about 1 ms for the ballast layer to be compressed and restored. Following compression and restoration of the ballast layer, sleeper-jumping occurs. The cause of this sleeper-jumping is the strain energy stored in the ballast under the

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**Figure 24** presents amplitude spectra of the vertical displacement of the sleeper. Results indicate that rigid-body natural vibration occurs at around 120 Hz. The value is approximately 10–20% larger than the experimental one and is almost one-third of the elastic natural vibration frequency. Large-scale finite element analysis by tensionless analysis reveals that rigid-body natural vibration mode occurs at almost one-third of the elastic natural vibration frequency.

**Figure 25** presents time history waveforms of response of the von Mises stress on the crosssection inside the angular part of the gravel where the maximum response stress occurs. This analysis shows that strong stresses up to 80 MPa can be observed in tetrahedral elements near the contact part, whereas the average maximum pressure on the ballast surface is 74 kPa. The stress acting on the angular part is approximately 1100 times greater than the average value of the loading stress on the ballast surface. Assuming that the unconfined compressive strength is 60 MPa at the angular part of the ballast gravel, the application of a dynamic load of 55 kPa or more to the surface of the ballast layer under the bottom surface of the sleeper would cause minute fracturing

**9. Stress on the cross-section inside the angular part of the gravel**

or breakage around the angular part of the ballast gravel, where the stress will converge.

**Figure 26** presents results of measurements indicating the maximum values for all sensors (measured in a cross-sectional square area with 8 cm sides on the whole bottom surface of the sleeper, total: 75 pieces, 25 units × 3 rows) as the passenger train passed. The figure also shows a threshold line of 55 kPa at which breakage will occur in the angular part of the ballast gravel.

compression procedure and its abrupt release.

**Figure 24.** Response spectrum of vertical displacement related to impulse loading.

**Figure 22.** Spectra of von Mises stress.

**Figure 23.** Time history response of sleeper displacement.

**Figure 24.** Response spectrum of vertical displacement related to impulse loading.

considerable stresses of approximately 57 MPa are obtained near the surface of the ballast layer under the sleeper bottom. In the figure, in general, the positive motion denotes compression and downward behaviour. Negative motion denotes tension and upward behaviour.

**Figure 22** displays the linear amplitude spectra of response of von Mises stress, on the crosssection inside the gravel angle acting at different depths below the loading point of the left rail, which are obtained by conducting fast Fourier transformation of these time history response waveforms and by smoothing them at 20 Hz bandwidth. As the figure shows, these spectral curves identify the first-order elastic vibration resonance mode of the ballast layer at around

**Figure 23** presents the time history of response of the vertical displacement at the left edge of the sleeper immediately after the 0.1 ms square impulse loading waveforms of 100 kN. In this figure, the downward displacement shows the downward motion of the sleeper (i.e. ballast

337 Hz, where the entire ballast layer stretches vertically as an elastic body.

**Figure 22.** Spectra of von Mises stress.

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**Figure 23.** Time history response of sleeper displacement.

layer compression); the upward displacement represents the upward motion of sleeper (i.e. ballast layer extension). According to the figure, when an impact load is applied, the ballast layer instantaneously deforms elastically because of compression. It then returns to the preloading position. Consequently, it takes only about 1 ms for the ballast layer to be compressed and restored. Following compression and restoration of the ballast layer, sleeper-jumping occurs. The cause of this sleeper-jumping is the strain energy stored in the ballast under the compression procedure and its abrupt release.

**Figure 24** presents amplitude spectra of the vertical displacement of the sleeper. Results indicate that rigid-body natural vibration occurs at around 120 Hz. The value is approximately 10–20% larger than the experimental one and is almost one-third of the elastic natural vibration frequency. Large-scale finite element analysis by tensionless analysis reveals that rigid-body natural vibration mode occurs at almost one-third of the elastic natural vibration frequency.

## **9. Stress on the cross-section inside the angular part of the gravel**

**Figure 25** presents time history waveforms of response of the von Mises stress on the crosssection inside the angular part of the gravel where the maximum response stress occurs. This analysis shows that strong stresses up to 80 MPa can be observed in tetrahedral elements near the contact part, whereas the average maximum pressure on the ballast surface is 74 kPa. The stress acting on the angular part is approximately 1100 times greater than the average value of the loading stress on the ballast surface. Assuming that the unconfined compressive strength is 60 MPa at the angular part of the ballast gravel, the application of a dynamic load of 55 kPa or more to the surface of the ballast layer under the bottom surface of the sleeper would cause minute fracturing or breakage around the angular part of the ballast gravel, where the stress will converge.

**Figure 26** presents results of measurements indicating the maximum values for all sensors (measured in a cross-sectional square area with 8 cm sides on the whole bottom surface of the sleeper, total: 75 pieces, 25 units × 3 rows) as the passenger train passed. The figure also shows a threshold line of 55 kPa at which breakage will occur in the angular part of the ballast gravel.

**Figure 25.** Time history response waveforms of the von Mises stress.

The analytical result coincides to a considerable degree with those obtained using in situ measurements and full-scale experiments. Accordingly, the natural mode which is expected to occur when an impact load is applied is determined mainly by the contact condition on

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**Figure 27.** Structural damping coefficient in the vertical direction for a 30 cm-thick ballast layer.

As shown in **Figure 5** presented earlier, the amplitude of displacement of the ballast gravel decreases in inverse proportion to the squares of the frequency, according to the physical theory. Therefore, the occurrence of the rigid-body natural vibration at one-third of the elastic natural vibration frequency is expected to induce nine times larger amplitude of displacement within the composite structure of the ballasted track than in the case of the elastic natural frequency. The occurrence of the rigid-body natural vibration is expected to contribute greatly to the progress of the ballast deterioration. In theory, improvement of the contact condition between the sleeper bottom and the boundary region of the ballast layer might reduce the amplitude of displacement to one-ninth at most compared with that in the current status of the ballasted track. Moreover, measurements of the ratio of the rigid-body natural vibration and the elastic vibration at the site

are expected to contribute to quantitative evaluation of the ballasted track condition.

**Figure 27** presents the structural damping coefficient on the sleeper bottom in the vertical direction for a 30 cm-thick ballast layer, as identified by the experimental model analysis using full-scale mock-ups of the ballasted track and precise finite element analysis according to an earlier report of the literature [24]. As the figure shows, the structural damping factor of the ballasted layer has extremely strong dependence on the frequency. The ballast layer, in the high-frequency domain over 200 Hz, provides extremely high damping functions for reducing the impact energy. However, the ballast layer is almost non-resistant to the wave components of dynamic loads in the low-frequency domain. The load components in the low-frequency domain will be reduced only slightly unless the ballast aggregate is fully constrained by an appropriate amount of uniform pressure from the surrounding area. To reduce the ballast degradation based on these mechanisms, improvement of the contact condition of the boundary region adjacent to the sleeper bottom in the ballast layer contributes to restraint of the occurrence of the rigid-body natural vibration modes at low frequencies.

the sleeper bottom.

**Figure 26.** Distribution of the maximum load acting during passage of a passenger train.

Minute breakage might occur at points within the ballast layer where the stress converges if the measured value exceeds this line. As the figure shows, this limit was exceeded in 40 out of the 75 sensors; the ratio of exceeding the threshold is 53%. The results indicate that the degradation of the ballast layer might occur at any time under the effects of regular train passage. Further experimentation and analysis must be conducted to clarify this issue.

#### **10. Relations among natural vibration modes of ballasted track**

According to the results, when information related to the contact points between the sleeper bottom and ballast pieces is modeled with MPCs, numerically obtained results show that the vertical elastic natural vibration mode of the ballast layer occurred at about 310 Hz. However, when the contact point is modeled with nonlinear contact springs, i.e. with tensionless contact, the rigid-body natural vibration mode is found numerically as approximately 120 Hz.

**Figure 27.** Structural damping coefficient in the vertical direction for a 30 cm-thick ballast layer.

Minute breakage might occur at points within the ballast layer where the stress converges if the measured value exceeds this line. As the figure shows, this limit was exceeded in 40 out of the 75 sensors; the ratio of exceeding the threshold is 53%. The results indicate that the degradation of the ballast layer might occur at any time under the effects of regular train passage.

According to the results, when information related to the contact points between the sleeper bottom and ballast pieces is modeled with MPCs, numerically obtained results show that the vertical elastic natural vibration mode of the ballast layer occurred at about 310 Hz. However, when the contact point is modeled with nonlinear contact springs, i.e. with tensionless contact, the rigid-body natural vibration mode is found numerically as approximately 120 Hz.

Further experimentation and analysis must be conducted to clarify this issue.

**Figure 26.** Distribution of the maximum load acting during passage of a passenger train.

**Figure 25.** Time history response waveforms of the von Mises stress.

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**10. Relations among natural vibration modes of ballasted track**

The analytical result coincides to a considerable degree with those obtained using in situ measurements and full-scale experiments. Accordingly, the natural mode which is expected to occur when an impact load is applied is determined mainly by the contact condition on the sleeper bottom.

As shown in **Figure 5** presented earlier, the amplitude of displacement of the ballast gravel decreases in inverse proportion to the squares of the frequency, according to the physical theory. Therefore, the occurrence of the rigid-body natural vibration at one-third of the elastic natural vibration frequency is expected to induce nine times larger amplitude of displacement within the composite structure of the ballasted track than in the case of the elastic natural frequency. The occurrence of the rigid-body natural vibration is expected to contribute greatly to the progress of the ballast deterioration. In theory, improvement of the contact condition between the sleeper bottom and the boundary region of the ballast layer might reduce the amplitude of displacement to one-ninth at most compared with that in the current status of the ballasted track. Moreover, measurements of the ratio of the rigid-body natural vibration and the elastic vibration at the site are expected to contribute to quantitative evaluation of the ballasted track condition.

**Figure 27** presents the structural damping coefficient on the sleeper bottom in the vertical direction for a 30 cm-thick ballast layer, as identified by the experimental model analysis using full-scale mock-ups of the ballasted track and precise finite element analysis according to an earlier report of the literature [24]. As the figure shows, the structural damping factor of the ballasted layer has extremely strong dependence on the frequency. The ballast layer, in the high-frequency domain over 200 Hz, provides extremely high damping functions for reducing the impact energy. However, the ballast layer is almost non-resistant to the wave components of dynamic loads in the low-frequency domain. The load components in the low-frequency domain will be reduced only slightly unless the ballast aggregate is fully constrained by an appropriate amount of uniform pressure from the surrounding area. To reduce the ballast degradation based on these mechanisms, improvement of the contact condition of the boundary region adjacent to the sleeper bottom in the ballast layer contributes to restraint of the occurrence of the rigid-body natural vibration modes at low frequencies.
