## 3 Dynamic Analysis
### Result of Dynamic Analyses
#### 3.3.3 Dynamic Response Attenuation in Soil  
The \(A_{max}\) commonly decreases with distance from source, and it seems that this general rule can also be found in the responses of track bed as shown in Figure 11. The acceleration levels along measurement line show significant decrease with increasing distance. The \(A_{max}\) commonly decreases with distance from source, and only approximately 30-56% remains after attenuation for a subgrade bed depth of 2.5 m, whose value can be remained around 5% after attenuation of 7.0 m subgrade. Furthermore, \(A_{max}\) tends to be uniform in the embankment layer at different \(v\) and \(F\), as the \(A_{max}\) values are nearly coincident at a soil depth of more than 7.0 m. Additionally, Figure 11 also indicates that \(v\) and \(F\) have significant effects on the Amax on the top layer surface. It increases with \(v\) and \(F\). However, this effect varies with depth from the source. The closer to the source, the greater the impact.  
In this work, \(A_{max}\) is normalized by the corresponding values obtained on the top layer surface and expressed as an attenuation ratio of vibration. As the attenuation ratio deviation between the different conditions is low, all the cases can be averaged, as shown in Figure 11b. A best-fit curve can exhibit the variation of the attenuation ratio profile with increase in the depth, which can be expressed as an exponential function:  
\[f_{AH}=1.07e^{-0.29H}-0.07~{}R^{2}=0.996 \tag{11}\]  
where \(f_{AH}\) is the attenuation ratio of the acceleration, \(e\) is the natural logarithm constant, and \(H\) is the distance to the bottom of the ballast along the track centerline.  
Figure 12a shows the vertical peak dynamic stress distribution (\(\sigma_{d,max}\)) against the depth, as generated by moving trains for different v and F. The train loads are translated through the top and bottom layers to the embankment in a wave form. In the process of translation, a certain amount of energy is absorbed by damping, and the translating area becomes wider. Similar to the attenuation characteristics of the acceleration, \(\sigma_{d,max}\) attenuates with an increase in the depth. However, Figure 12a indicates that \(\sigma_{d,max}\) attenuates considerably faster than Amax does. For example, at the bottom of the top layer (0.6 m), \(\sigma_{d,max}\) is attenuated by approximately 75%, whereas the attenuation is approximately 95% at the bottom of the bottom layer (2.5 m). \(\sigma_{d,max}\) is attenuated by nearly 98% at the foundation (8.1 m), resulting in a negligible value compared to the weight of the subgrade filling.  
Figure 10: Variation of vertical peak acceleration and dynamic stress at the center of the top layer surface with the axle load.  
A large proportion of \(\sigma_{d,max}\) is visibly attenuated in the top and bottom layers, and this result is consistent with that reported in existing literature [35].  
Figure 11: Acceleration distribution along the depth in the centerline for (**a**) vertical peak accelerations, \(A_{max}\), and (**b**) attenuation ratio,\(f_{AH}\).  
The results of normalization of \(\sigma_{d,max}\) at different depths using \(\sigma_{d,max}\) at the top layer surface are shown in Figure 12b. It can be noted that the data can be fitted well by using the proposed empirical formula, that is Equation (12), which indicates that the vertical attenuation laws of attenuation for different F are similar.  
\[f_{SH}=\frac{0.18}{0.18+H} \tag{12}\]  
where \(f_{SH}\) is the attenuation ratio of the dynamic stress.  
Figure 12: Acceleration distribution along the depth in the centerline for (**a**) vertical peak dynamic stress \(\sigma_{d,\,max}\) and (**b**) attenuation ratio (\(f_{SH}\)).