# Deformation Behavior of Saturated Soft Clay under Cyclic Loading with Principal Stress Rotation
## 4 Test Results and Discussion
### Backbone Curve
In order to study the deformation characteristics of soft clay under traffic loading with principal stress axis rotations, the torsional cyclic stress ratio \(\eta\) and cycles \(N\) were fitted to the parameters \(a\) and \(b\), and it was found that \(a\) and \(b\) simultaneously satisfy the following relation:  
\[a=p_{1}+p_{2}*\eta^{3}+p_{3}*\ln(N) \tag{11}\]  
where \(p_{1}=0.00098\), \(p_{2}=0.02103\), and \(p_{3}=0.00024\);  
\[b=p_{4}+p_{5}*\eta^{3}+p_{6}*\ln(N) \tag{12}\]  
where \(p_{4}=0.01508\), \(p_{5}=-0.00725\), and \(p_{6}=0.00143\)  
Substituting Equations (11) and (12) into Equation (10), a modified H-D backbone curve model considering cycles and principal stress rotations can be developed as follows:  
Figure 11:0  
To verify this model, the results for three cyclic tests with VCSR = 0.15 were used. Figure 14 shows the results of this H-D backbone curve model compared with the measured test data, and the predicted results are in good agreement with the measured data. The results show that the predicted value is larger than the measured value. The deviation between the predicted value and the measured value is less than 14% when the principal stress rotation is taken into account, while the deviation between the predicted value and the measured value is as high as 42% if the principal stress axis rotation is not taken into account. This modified model improves the prediction accuracy by a factor of three, demonstrating the accuracy of our model and the need to account for principal stress axis rotation for traffic loading. In the actual engineering, each structure must achieve a certain safety factor; the predicted values are on the high side to better meet this need for practical engineering and to ensure engineering safety. The real stress conditions in the pavement are much more complicated. This modified model can provide a preliminary reference for pavement design. Compared with the conventional H-D backbone curve model, the model constructed in the paper considers the effects of both the number of cycle turns and the torsional cyclic stress ratio \(\eta\). As a result, the model can better represent the stiffness degradation characteristics of soft clay under traffic loading and is more realistic. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{**No. of Cycles, \(N\)**} & **Parameter, \(a\)** & **Parameter, \(b\)** & **R-Squared, \(\mathbf{R}^{2}\)** & \multicolumn{1}{c}{**Cyclic Shear Stress**} \\  & & & & **Ratio, \(\eta\)** \\ \hline
10 & 0.00101 & 0.0217 & 0.9462 & \\
100 & 0.00160 & 0.0211 & 0.9717 & \\
500 & 0.00265 & 0.0204 & 0.8441 & \\
1000 & 0.00439 & 0.0184 & 0.9717 & \\ \hline
10 & 0.00199 & 0.0177 & 0.9890 & \\
100 & 0.00254 & 0.0192 & 0.9883 & \\
500 & 0.00291 & 0.0207 & 0.9761 & \\
1000 & 0.00318 & 0.0210 & 0.9647 & \\ \hline
10 & 0.00463 & 0.0133 & 0.9860 & \\
100 & 0.00498 & 0.0207 & 0.9487 & \\
500 & 0.00524 & 0.0244 & 0.9350 & \\
1000 & 0.00543 & 0.0253 & 0.9338 & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fitting parameters.