# Evaluation of Aeolian Sand Collapsibility Based on Physical Indicators in the Mu Us Desert, China
## 5 Construction of Collapsibility Evaluation Method
\[N/D_{r}\textsuperscript{2}=177.1921-5.97565\sigma_{v}^{{}^{\prime}} \tag{5}\]  
To obtain the relationship between \(N\) and \(\rho_{d}\) directly, \(D_{r}\) was replaced by the equivalent equation, and Equation (4) was substituted into Equation (5), transforming it as follows: \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Test Pit Number** &
\begin{tabular}{c} **Number of Hammer** \\ **Blows (N)** \\ \end{tabular} & **Effective Overlying Stress (\(\sigma_{v}^{{}^{\prime}}\)/kPa)** & **Relative Density (\(D_{r}\)/\(\%\))** & \(\frac{N}{D_{r}\textsuperscript{2}}\) \\ \hline \hline \(\mathrm{JSZH}\) 1-1 & 7 & 23.51 & 44 & 36.137 \\ \(\mathrm{JSZH}\) 1-2 & 8 & 23.58 & 46 & 37.355 \\ \(\mathrm{JSZH}\) 2-1 & 8 & 24.00 & 49 & 33.548 \\ \(\mathrm{JSZH}\) 2-2 & 9 & 24.21 & 51 & 34.748 \\ \(\mathrm{JSZH}\) 3-1 & 10 & 24.32 & 56 & 31.492 \\ \(\mathrm{JSZH}\) 3-2 & 11 & 24.29 & 59 & 31.647 \\ \(\mathrm{JSZH}\) 4-1 & 6 & 23.81 & 41 & 34.942 \\ \(\mathrm{JSZH}\) 5-1 & 5 & 23.82 & 39 & 32.952 \\ \(\mathrm{JSZH}\) 5-2 & 6 & 24.02 & 42 & 34.256 \\ \(\mathrm{JSZH}\) 5-3 & 9 & 24.14 & 52 & 33.284 \\ \(\mathrm{JSZH}\) 6-1 & 11 & 24.90 & 63 & 27.933 \\ \(\mathrm{JSZH}\) 6-2 & 11 & 24.90 & 63 & 27.539 \\ \(\mathrm{JSZH}\) 6-3 & 12 & 25.05 & 66 & 27.529 \\ \(\mathrm{JSZH}\) 6-4 & 12 & 25.05 & 65 & 28.173 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Values of each variable in Equation (3). \[\frac{N}{177.1921-5.97565\sigma_{v}^{{}^{\prime}}}=\left(\frac{(\rho_{d}-\rho_{dmin} )\rho_{dmax}}{(\rho_{dmax}-\rho_{dmin})\rho_{d}}\right)^{2} \tag{6}\]  
Considering the influence of the hydraulic head in sand, the effective overlying stress (\(\sigma_{v}^{{}^{\prime}}\)) can be converted to the following:  
\[\sigma_{v}^{{}^{\prime}}=(\gamma_{1}-\gamma_{w})h_{w}+\gamma_{2}(h-h_{w})=( \rho_{1}-\rho_{w})gh_{w}+\rho_{2}(h-h_{w})g \tag{7}\]  
where \(\gamma_{1}\) is the gravity density of the soil under the water level, \(\gamma_{w}\) is the gravity density of the water, \(h_{w}\) is the distance from the water level to the target test soil layer, \(\gamma_{2}\) is the gravity density of the soil above the water level, \(h\) is the depth of the target test soil layer from the ground, \(\rho_{1}\) is the density of the soil under the water level, \(\rho_{w}\) is the density of water (constant), \(g\) is the acceleration of gravity, and \(\rho_{2}\) is the density of the soil above the water level. The conversion formula to relate the density to the dry density and water content is as follows:  
\[\rho=\rho_{d}(1+w) \tag{8}\]  
where \(w\) is the water content of the soil sample. By substituting Equations (7) and (8) into Equation (6), Equation (6) can be transformed into the following:  
\[\frac{N}{177.1921-5.97565[(\rho_{d}(1+w_{1})-\rho_{w})gh_{w}+\rho_{d}(1+w_{2}) (h-h_{w})g]}=\left(\frac{(\rho_{d}-\rho_{dmin})\rho_{dmax}}{(\rho_{dmax}-\rho _{dmin})\rho_{d}}\right)^{2} \tag{9}\]  
where \(w_{1}\) is the water content of the soil below the water level, \(w_{2}\) is the water content of the soil above the water level, and the other parameters are explained in Equations (6)-(8). In summary, after establishing the relationships in Equation (9), we needed to conduct standard penetration tests and indoor geotechnical tests based on disturbance samples of the target test soil layer. After obtaining the number of hammer blows (\(N\)), the distance from the water level to the target test soil layer (\(h_{w}\)), the depth of the target test soil layer from the ground (\(h\)), the maximum dry density (\(\rho_{dmax}\)), the minimum dry density (\(\rho_{dmin}\)), the water content of the soil under the water level (\(w_{1}\)), and the water content of the soil above the water level (\(w_{2}\)) through formula calculations, we could obtain the dry density (\(\rho_{d}\)) through Equation (9).