# Influence of Different Freezing Modes on the Base Slab Displacement of an Upper Structure
## 2 Similarity Simulation Experiment
### Modeling Test  
As is well known, a correct physical model must satisfy a series of similarity laws that consist of temperature, humidity, stress and displacement. By using the AGF method, the physical process of a tunnel underpassing upper structures can be divided into the following three stages. (i) Chilled brine extracts heat from soil, and then, pore water is frozen to ice. The similarity principle of a temperature field [48; 49] can be represented as:  
\[F\left(\frac{Q}{cT},\frac{at}{r_{0}^{2}},\frac{T_{y}}{T_{0}},\frac{T_{c}}{T_{ 0}},\frac{r_{f}}{r_{0}}\right)=0 \tag{1}\]  
where \(Q\) is the latent heat of soil, \(c\) is the specific heat of soil, \(T\) is temperature. \(a\) is the thermal diffusivity of soil, \(t\) is time, \(r_{0}\) is a radial coordinate, \(r_{f}\) is the outer radius of a freezing pipe, \(T_{0}\) is the initial temperature, \(T_{y}\) is the freezing point of soil, and \(T_{c}\) is the surface temperature of a freezing pipe [50]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & **Density** & **Water** & **Saturation** & **Freezing** & **Specific Heat Capacity** \\  & **(g cm\({}^{-3}\))** & **Content (\%)** & **(\%)** & **Temperature (\({}^{\circ}\)C)** & **(W-m\({}^{-1,\circ}\)C\({}^{-1}\))** \\ \hline Clay & 1.78 & 38.5 & 97 & 0.1 & 1.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The physical and mechanical parameters of grey silty clay. Figure 1: Schematic illustration of a closely underpassing project in Shanghai, China. The testing soil was collected from the site. Then, the compactness and water content of soil were adjusted to be consistent with the undisturbed soil. The experimental temperature was also chosen as the in-situ one. Therefore, other parameters such as the internal friction angle, Poisson's ratio, and void ratio can be considered the same as the in-situ soil. The similarity principle of a humidity field can be written as:  
\[\Theta=\frac{w}{w_{0}} \tag{2}\]  
where \(w\) is the initial water content of soil and \(w_{0}\) is the unfrozen water content of frozen soil. Because the process of water migration is similar to thermal conduction [49], the similarity ratio of a humidity field is approximately equal to that of a temperature field. (ii) Freezing and excavation cause the deformation of soil. Here, the stress-displacement field can be expressed in a dimensionless form [49; 51], that is:  
\[F\left(\frac{\sigma}{E},\frac{P}{E},\frac{P}{rH},\frac{d}{u},v\right)=0 \tag{3}\]  
where \(\sigma\) is stress, \(E\) is the elastic modulus, \(P\) is load, \(r\) is the bulk density, \(H\) is the tunnel depth, \(d\) is the thickness of frozen soil, \(u\) is displacement, and \(v\) is Poisson's ratio of soil. (iii) The upper structure is deformed by frost heave or settlement. In terms of the linear elastic mechanics, deflection is inversely proportional to stiffness, and we have:  
\[u_{s\propto 1}/K, \tag{4}\]  
where \(u_{s}\) is the displacement of a base slab causing by the frost force and \(K=EI\) is the stiffness of upper structures, with \(I\) the static moment of a base slab. Based on these similarity laws, the similarity ratios of relevant parameters can be represented as:  
\[ \begin{split} C_{T}=C_{s}=1,\\ C_{t}=n^{2},\\ C_{L}=C_{p}=C_{d}=n.\end{split}  \tag{5}\]  
where \(n\) is a constant, \(C_{T}\), \(C_{s}\), \(C_{t}\), \(C_{L}\), \(C_{p}\) and \(C_{d}\) are the similarity ratios of temperature, stress, time, length, density and displacement, respectively. Taking the boundaries of the station and frozen walls, the size of a laboratory and the operability of a model into account, the geometric similarity ratio \(C_{L}\) was set to be \(n=1/25\). Then, a steel chamber was established with a length of 3 m, a height of 1.5 m, and a width of 1.5 m on a three-dimensional platform, as shown in Figure 2a. Figure 2: (**a**) The in-situ experimental photograph and (**b**) its schematic diagram (in the unit of mm). To do testing, a soil layer was packed in the steel chamber.