## 4 Discussion
### Comprehensive Rate Evaluation Coefficient \(C\) Against Cyclic Amplitude  
The comprehensive evaluation coefficient \(C\) can be plotted against cyclic loading amplitude, as shown in Figure 15. For a given model with a certain compaction degree, the coefficient \(C\) is increased at an accelerated rate with cyclic amplitude \(P_{d}\) increasing. Meanwhile, at a given cyclic amplitude, \(C\) is decreased as compaction degree \(K\) becomes larger. As mentioned herein before, for the three testing models at different compaction degrees, ground coefficient \(K_{30}\) has been evaluated. Furthermore, it has been proposed by [PERSON] [23] that there is an empirical relationship between ground coefficient \(K_{30}\) and load bearing capacity \(P_{cr}\), as shown by the following equation:  
\[P_{cr}=2.4K_{30}+15 \tag{7}\]  
Then, bearing capacity \(P_{cr}\) for the three testing models can be quantitatively evaluated with Equation (7) as a reference. The values of \(P_{cr}\) are 341.4 kPa, 552.6 kPa, and 910.2 kPa for the models with compaction degrees of 0.9, 0.95, and 1.0 respectively. Obviously, there is a positive relationship  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(P_{d}\) (kPa)} & \multicolumn{2}{c}{\(K\) = 0.9} & \multicolumn{2}{c}{\(K\) = 0.95} & \multicolumn{2}{c}{\(K\) = 1.0} \\ \cline{2-7}  & \(C\) & \(m\) & \(C\) & \(m\) & \(C\) & \(m\) \\ \hline
50 & 0.017 & 1.119 & 0.014 & 1.287 & 0.005 & 1.555 \\
100 & 0.159 & 0.708 & 0.054 & 0.843 & 0.023 & 1.084 \\
150 & 0.524 & 0.661 & 0.115 & 0.701 & 0.051 & 0.804 \\
200 & 1.348 & 0.636 & 0.214 & 0.646 & 0.096 & 0.652 \\
250 & 2.489 & 0.550 & 0.295 & 0.594 & 0.139 & 0.602 \\
300 & 7.026 & 0.299 & 0.400 & 0.545 & 0.178 & 0.559 \\
350 & & & & 0.730 & 0.498 & 0.220 & 0.515 \\
400 & & & & 1.104 & 0.436 & 0.273 & 0.481 \\
500 & & & & 3.647 & 0.050 & 0.527 & 0.394 \\
600 & & & & & 0.910 & 0.217 \\
700 & & & & & 2.243 & 0.101 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The fitting value of \(C\) and \(m\) for the three models.  
between bearing capacity \(P_{cr}\) and compaction degree \(K\). Therefore, it can be deduced that coefficient \(C\) is negatively correlated with bearing capacity \(P_{cr}\) at a given amplitude of cyclic loading.  
To demonstrate the inner connection between \(C\) and \(P_{cr}\) as well as cyclic amplitude \(P_{d}\), comprehensive evaluation coefficient \(C\) is further plotted against \(P_{d}/P_{cr}\), as shown in Figure 16. It is interesting to find that the parameter \(C\) under different cyclic loading amplitudes for the three models reasonably develops along a unique curve. As the ratio \(P_{d}/P_{cr}\) increases, the comprehensive evaluation coefficient \(C\) also increases exponentially. A regression analysis is performed and gives the following equation a correlation coefficient of 0.85:  
\[C=0.03\bigg{(}e^{5.83\frac{P_{d}}{P_{cr}}}-1\bigg{)} \tag{8}\]  
The fitting curve is also plotted in Figure 16. It can be seen that the testing data of the parameter \(C\) locates closely along the fitting curve, which demonstrates that Equation (8) can be adopted to evaluate the comprehensive evaluation coefficient \(C\). It is worth recalling that the parameter \(C\) is mentioned above as an indicator of average deformation rate for a given amplitude of cyclic loading. Given the expression of Equation (8), it is remarkable that the average deformation rate increases with the amplitudes of cyclic loading, and decreases with bearing capacity \(P_{cr}\) as well as compaction degree \(K\) increasing. This conclusion is in accordance with our common sense.  
Figure 16: The relationship between \(C\) and \(P_{d}/P_{cr}\).  
Figure 15: Comprehensive rate evaluation coefficient against cyclic amplitude.