# Evaluation of Aeolian Sand Collapsibility Based on Physical Indicators in the Mu Us Desert, China
## 5 Construction of Collapsibility Evaluation Method  
Relationship Between the Number of Hammer Blows in the Standard Penetration Test and the Dry Density of Aeolian Sand  
As a physical property indicator in this evaluation method, the dry density can be represented by the number of hammer blows recorded in a standard penetration test (SPT). During the representation process, an intermediate parameter of relative density must be introduced. [PERSON] et al. [49] highlighted three main principles for the application of relative density. First, the clay content of the soil must be less than 15%. Second, if the diameter of the gravel in the soil is as large as 75 mm, 70% of the soil particles must be less than 37.5 mm in diameter. Third, the soil should be non-sticky and drain smoothly. The aeolian sands in the Mu Us Desert meet these three conditions. 1.1 Relationship for \(N_{60}\)-\(D_{r}\)-\(\sigma_{v}^{{}^{\prime}}\) in Normally Consolidated Sandy Soil  
After conducting a substantial number of experiments on naturally deposited and normally consolidated sand, [PERSON] [50] showed that the empirical relationship between \(N_{60}\)-\(D_{r}\)-\(\sigma_{v}^{{}^{\prime}}\) can be expressed as follows:  
\[N_{60}/D_{r}{}^{2}=a+b\sigma_{v}^{{}^{\prime}} \tag{1}\]  
where \(D_{r}\) is the relative density of the soil, \(\sigma_{v}^{{}^{\prime}}\) is the effective overlying stress, \(N_{60}\) is the number of hammer blows after normalization, and \(a\) and \(b\) are undetermined coefficients. [PERSON] et al. [51] proposed \(N_{60}\), which indicates that the measured \(N\) of different standard penetration test equipment was uniformly corrected to the standard hammer number, \(N_{60}\). The standardized formula is as follows:  
\[N_{60}=C_{ER}\cdot N \tag{2}\]  
where \(C_{ER}\) is the coefficient of correcting energy and \(N\) is the unnormalized number of hammer blows. Considering that the standard penetration tests are automatically decoupled, previous studies have reported [52] that \(C_{ER}=1\); thus, \(N_{60}=N\). Finally, Equation (1) can be transformed into the following:  
\[N/D_{r}{}^{2}=a+b\sigma_{v}^{{}^{\prime}} \tag{3}\]  
The relative density (\(D_{r}\)) in Equation (3) can be calculated using Equation (4); the results are presented in Table 4:  
\[D_{r}=\frac{(\rho_{d}-\rho_{dmin})\rho_{dmax}}{(\rho_{dmax}-\rho_{dmin})\rho_ {d}} \tag{4}\]  
where \(\rho_{d}\) is the dry density of the soil sample, \(\rho_{dmin}\) is the minimum dry density of the soil sample, and \(\rho_{dmax}\) is the maximum dry density of the soil sample. 1.2 Empirical Relationship of N-\(\rho_{d}\)-\(\omega\) for Aeolian Sand on the Southern Edge of the Mu Us Desert  
Equation (3) links the number of hammer blows (\(N\)), the relative density (\(D_{r}\)), and the effective overlying stress (\(\sigma_{v}^{{}^{\prime}}\)). Among them, the value of \(N\) was obtained from the in situ standard penetration test, whose results are listed in Table 2. The maximum dry density (\(\rho_{dmax}\)), minimum dry density (\(\rho_{dmin}\)), and dry density (\(\rho_{d}\)) in Equation (4) were obtained from the indoor geotechnical test, whose results are listed in Table 3. The effective overlying stress (\(\sigma_{v}^{{}^{\prime}}\)) was calculated according to Equation (7), whose results are listed in Table 4. The density value in Equation (7) was obtained from the indoor geotechnical tests. Next, the values of parameters a and b in Equation (3) were obtained by fitting the test results. Equation (3) shows that the effective overlying stress (\(\sigma_{v}^{{}^{\prime}}\)) has a linear relationship with the parameter (\(N/D_{r}\)2). Based on the data in Table 4, a = 177.1921, b = \(-\)5.97565, and \(R^{2}\) = 0.90662 were obtained through fitting (via the Origin 2022 software). The fitting effect was strong, as shown by the results in Figure 9. The fitting equation is as follows:  
Footnote 2: The effective overlying stress (\(\sigma_{v}^{{}^{\prime}}\) / kPa) is calculated according to Equation (7).