# Durability-Aimed Design Criteria of Cement-Stabilized Loess Subgrade for Railway
## 4 The Design Criteria Aiming at Subgrade Durability  
### Determination of Dynamic Stress  
The strength control method is usually used to determine the thickness of the railway subgrade bed and material strength, which is the maximum dynamic stress \(\sigma_{\text{dmax}}\) transferred to filling through the subgrade bed under the train load must be less than the allowable strength of filling. As for measured values of dynamic stress, there is a belief that that the dynamic response of the existing railway subgrade was 35.0-86.0 kPa in China [63]. In addition, the dynamic stress of some specific test sections of the railway was specially tested by some Chinese scholars [64, 65, 66, 27, 31, 67]. The study showed that the dynamic stress of subgrade surface does not change significantly with train speed but increases with the increase of axle load [68]. As for the calculated value of dynamic stress, the formula of the maximum dynamic stress \(\sigma_{\text{max}}\) of subgrade surface was given in \"Code for Design of Intercity Railway\" [69] and \"Code for Design of High Speed Railway\" [70]. In addition to the criteria, some scholars calculated the dynamic stress by finite element simulation [31, 32]. It was found that the calculated values are in good agreement with the measured values. It can verify the possibility of finite element analysis. The measured and theoretical data showed that the dynamic stress of the subgrade decreases rapidly with depth [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. As for the  
Figure 3: The relationship between \(\eta_{\text{D}}\) amd \(N\). reduction law of dynamic stress along the depth, it can be known that the calculated value of dynamic stress of subgrade is generally greater than the measured value according to the above research results. Based on the existing research results, it was determined that the dynamic stress of the subgrade bottom is 20-50 kPa, and the dynamic stress of embankment below the subgrade bottom is 10-25 kPa under a certain guaranteed rate. As for the critical dynamic strength of filling, when the critical dynamic strength of subgrade filling is less than the dynamic stress generated by the dynamic train load, the soil structure will be destabilized, and the deformation will increase rapidly. However, when the situation is opposite, the soil is gradually compacted by load, and the increment of strain decreases gradually. If the load reaches a certain time, the soil density reaches a certain degree. Therefore, the actual dynamic stress is less than the critical dynamic strength, and the dynamic stress of the subgrade bed is stable for a long time. It was found that the minimum value of dynamic strength is 30 kPa even for the fine-grain soil, which is sensitive to water [31; 71; 72]. Additionally, the calculated method of the critical dynamic strength [\(\sigma_{\text{d}}\)] using foundation coefficient \(K_{30}\) or critical static strength [\(\sigma_{0}\)] was obtained [32] by analyzing the field data of subgrade of Da-Qin Line, which is shown in Equation (13) and Table 8. It can be seen that the calculated value is basically consistent with the measured value. According to the respective requirements of \(K_{30}\) for the filling at the subgrade bottom and the embankment below the subgrade bottom in \"Code for Design of Railway Earth Structure\" [12], the critical dynamic strength of filling at the subgrade bottom and the embankment below the subgrade bottom is, respectively, 147 kPa and 104 kPa. \[[\sigma_{\text{d}}]=0.45[\sigma_{0}]=1.08K_{30}+6.75 \tag{13}\]  
The above analysis shows that the critical dynamic strength of subgrade filling is greater than the dynamic stress transferred to subgrade by the dynamic load. If the strength reduction coefficient of subgrade filling is determined, the dynamic stress in the subgrade structure layers needs to be determined. As is known, the dynamic stress of the subgrade bottom and embankment below the subgrade bottom is not fixed.