## 6 Conclusions  
In this study, a comparison of the shear modulus measurements in cyclic triaxial tests using various determination methods is performed. The sand specimens, prepared with three relative densities, are tested at medium to large strains and from low to high confining pressures. Four methods, namely: hysteresis loop method, least square method, autocorrelation function method, and arithmetical average method, are adopted to determine the shear modulus from the second hysteresis loop. The uncertainty associated with the determination method is quantified. Based on the comparative study performed, the following conclusions can be drawn:  
Figure 16: \(G/G_{\text{max}}\) curves of two determination methods for \(N=2\), \(3\), \(4\) and the corresponding percentage changes.  
1. The hysteresis loop is generally symmetrical and it becomes progressively asymmetric with increasing \(\gamma\). In addition, the loading cyclic number N has a pronounced influence on the hysteresis loops with the influence becoming larger as \(\gamma\) increases. Both the \(G\) and \(G/G_{\rm max}\) show a tendency of decrease with increasing N.
2. The \(G\) and \(G/G_{\rm max}\) are strongly dependent on amplitudes of \(\gamma\) and \(\sigma_{\rm c}\). The increase in \(\sigma_{\rm c}\) leads to an increase in the \(G\) and \(G/G_{\rm max}\) whereas the \(G\) decreases as \(\gamma\) increases. At medium shear strains (10\(-\)4\(-\)10\(-\)3), the variation in \(G\) and \(G/G_{\rm max}\) is not significant, with a percentage change generally less than 5%. However, the difference increases with decreasing \(\sigma_{\rm c}\) or increasing \(\gamma\), with the maximum percentage change of around 30% at \(\sigma_{\rm c}\) = 50 kPa.
3. The arithmetical average method predicts almost identical results with the hysteresis loop method. Nevertheless, the larger \(G\) estimated at medium strains and the smaller \(G\) estimated at large strains for the autocorrelation function method, and least square method results in a steeper gradient of the \(G/G_{\rm max}\) curve. This thus corresponds to a smaller reference strain for these two methods.
4. Concerning the definitions of \(G_{\rm max}\), a relatively large \(G_{\rm max}\) is calculated by 1/a, leading to an increased nonlinearity of \(G/G_{\rm max}\). The \(G/G_{\rm max}\) determined by the autocorrelation function method is typically 30\(-\)40% lower than that of the hysteresis loop method in the strain range of 10\(-\)3\(-\)10\(-\)2 at \(\sigma_{\rm c}\) = 50 kPa while the percentage change is slightly small (i.e., 20\(-\)25%) for the \(G_{\gamma=10}-6\) case.
5. The autocorrelation function method has no capacity of capturing the dependency of \(G\) and \(G/G_{\rm max}\) on N. Both the hysteresis loop method and the least square method indicate an increased nonlinearity for high N. The comparison further demonstrates the influence of N on the uncertainty quantification of determination methods, particularly for large strains.  
Conceptualization, H.L.; Data curation, D.S. and Q.S.; Funding acquisition, H.L.; Formal analysis, Q.S. and D.S.; Investigation, Q.S. and D.S.; Methodology, Q.S. and D.S.; Supervision, H.L. and Q.S.; Validation, H.L. and Q.S.; Visualization, H.L. and Q.S.; Writing--original draft, Q.S.; Writing--review and editing, Q.S. All authors have read and agreed to the published version of the manuscript.  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
Not applicable.  
The authors declare no conflict of interest.