# A Method to Estimate Dynamic Pore Water Pressure Growth of Saturated Sand-Gravel Materials
## 2 Test Design
### Test Conditions  
For this study, sand-gravel materials with a relative density \(D_{\rm r}\) of 0.9 were consolidated under pressures of 50, 100, 200, and 300 kPa. A sine wave with a loading frequency of 0.33 Hz was employed to apply dynamic stress through vibration until failure occurred. Failure was defined as either an axial strain of 5% or excess pore water pressure reaching the confining pressure. This was performed to study the dynamic characteristics of the sand-gravel materials under different initial effective confining pressures \(\sigma_{\rm 3c}\), consolidation stress ratios \(K_{\rm c}\), and stress levels. Table 3 lists static and dynamic stress control conditions of this test. In dynamic triaxial tests, initial dynamic stress values are determined based on empirical experience. Once the samples meet the failure criteria, \(\sigma_{\rm d}\) for each condition \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Sand-Gravel** \\ **Materials in** \\ **Xinjiang** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Mean Grain Size** \\ **\(d_{\rm 30}\)/mm** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Specific Gravity** \\ **\(G_{\rm s}\)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Minimum Dry** \\ **Density** \\ **\(\rho_{\rm min}\)/g-cm\({}^{-3}\)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Maximum Dry** \\ **Density** \\ **\(\rho_{\rm max}\)/g-cm\({}^{-3}\)** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Relative Density** \\ **Density** \\ **\(D_{\rm r}\)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Dry Density** \\ **\(\rho_{\rm a}\)/g-cm\({}^{-3}\)** \\ \end{tabular} } \\ \cline{1-1}  & 14 & 2.75 & 1.85 & 2.34 & 0.9 & 2.28 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Physical properties of tested sand-gravel materials. Figure 3: Grain-size distribution of tested sand-gravel materials. Figure 2: GCTS dynamic triaxial test apparatus. is recorded in Table 3. In general engineering practice, the minimum principal stress of dam materials is estimated based on the dam height, and the maximum experimental confining pressure is set to exceed this value. This research aims to investigate the dynamic pore pressure of sand-gravel materials under low confining pressures; therefore, confining pressures \(\sigma_{\text{3c}}\) are set at 50, 100, 200, and 300 kPa. Most natural soils are in an anisotropic consolidation state; thus, soil strength parameters derived from such conditions are more reflective of real-world scenarios. Accordingly, the consolidation stress ratios, \(K_{\text{c}}\), are set at 1.5 and 2.0. Undrained cyclic triaxial tests were carried out under different cyclic stress ratios (CSRs) after consolidation. The CSR is the ratio of the dynamic shear stress on the 45\({}^{\circ}\) slope of the triaxial specimen to its initial normal stress; the formulas are as follows:  
\[\left\{ \begin{array}{l}K_{\text{c}}=\frac{\sigma_{\text{1c}}}{\sigma_{\text{ 3c}}}\\ \sigma_{0}^{\prime}=\frac{1+K_{\text{c}}}{2}\sigma_{\text{3c}}\\ \tau_{\text{d}}=\frac{\sigma_{\text{d}}}{2}\\ \text{CSR}=\frac{\tau_{\text{d}}}{\sigma_{0}^{\prime}}=\frac{\sigma_{\text{d} }}{(1+K_{\text{c}})\sigma_{\text{3c}}}\end{array} \right. \tag{1}\]  
where \(\sigma_{\text{1c}}\) is axial stress, \(\sigma_{\text{3c}}\) is the initial effective confining pressure, \(K_{\text{c}}\) is the initial consolidation stress ratio, \(\sigma_{0}^{\prime}\) is the initial normal stress, \(\tau_{\text{d}}\) represents the dynamic shear stress, \(\sigma_{\text{d}}\) is the axial cyclic stress, and CSR is cyclic stress ratio. During the test, the sample is consolidated under the stresses \(\sigma_{\text{1c}}\) and \(\sigma_{\text{3c}}\). At this point, the initial normal stress, \(\sigma_{0}^{\prime}\), on the sample's 45\({}^{\circ}\) slope is \((\sigma_{\text{1c}}+\sigma_{\text{3c}})/2\), and the initial shear stress is \((\sigma_{\text{1c}}-\sigma_{\text{3c}})/2\).