# Deformation Law and Control Limit of CRTSIII Slab Track under Subgrade Frost Heave
## 3 Mapping the Relationship of Track Irregularity under Frost Heave  
### Effect of Frost Heave Position  
The track irregularity caused by foundation displacement is usually considered as a cosine curve [52]. Based on the track irregularity test data [39] in the subgrade frost heave zone of the Harbin-Dalian high-speed railway in February 2013, the cosine function is adopted to simulate the subgrade frost heave displacement. The frost heave action positions are singular in this study. Using the displacement module [47] of the ABAQUS  
Figure 6: Diagram of concrete specimen failure modes. (**a**) Failure mode of C40 concrete specimen; (**b**) failure mode of C60 concrete specimen; (**c**) simulation and theoretical results of C40 concrete; (**d**) simulation and theoretical results of C60 concrete. software, the displacement function is applied to the nodes on the surface of the subgrade bed, which is a cosine function fitting the displacements of subgrade frost heave as shown in Figure 7a. The analysis type of the model is static analysis. Considering the cosine-type displacement curve of subgrade frost heave with a frost heave amplitude of 10 mm and a wavelength of 10 m, the effect of frost heave on the structure and mechanical properties at different positions of the base plate has been analyzed. Specifically, A is located at the seam of the base plate, B and D are at 1/4 and 3/4 of the track slab, C is at the middle of the track slab and E is at the middle of the base plate, as shown in Figure 7b. As shown in Figure 8, subgrade frost heave affects the deformation of its upper structure. Due to the greater rigidity of the CRTSIII slab track, a certain deformation deviation appears between the rail and subgrade. The rail's deformation is asymmetrical compared to the cosine curve, and the deformation amplitude from the subgrade mapped to the rail increases. In order to clarify the influence of subgrade frost heave on track irregularity, the ratio of peak value of rail displacement to the corresponding frost heave amplitude under frost heave displacement is defined as the amplitude transfer ratio. As shown in Figure 9, when frost heave occurs at position C (in the middle of the track slab), the amplitude transfer ratio is less than 1.0. Due to the large rigidity of pre-stressed track slab, the structural deformation is basically consistent with the frost heave curve, and no asymmetry occurs in the longitudinal direction, thereby weakening the rail's upward deformation trend. When frost heave occurs at other positions, the amplitude transfer ratio is almost the same, which is around at 1.06. In order to reflect the variation trends of track irregularities, the first-order derivation of the rail deformation curve along the longitudinal direction has been carried out to reflect the track irregularity change rate, as shown in Equation (12). \[\textit{ff}=\frac{d_{y}}{d_{x}} \tag{12}\]  
where \(\alpha\) is the track irregularity change rate, \(y\) is the rail's deformation amplitude under frost heave (mm), and \(x\) is the longitudinal position coordinate of railway line (mm). The track irregularity change rate under different frost heave positions has been obtained, as shown in Figure 10.  
Figure 7: Frost heave type and positions. (**a**) Cosine-type displacement curve of subgrade frost heave; (**b**) frost heave action positions. As shown in Figure 10, when frost heave occurs at positions A and E, the rail's deformation curve is completely symmetrical, and the maximum positive and negative change rates of the track irregularities are (\(\pm\)2.33) and (\(\pm\)2.26), respectively. When frost heave occurs at positions B and D, the rail's deformation curve is not symmetrical; similarly, the positive and negative change rates of the track irregularities are quite different. This is because the track slab's end warps under the action of frost heave, which leads to the rail's asymmetric deformation.