# Analysis of the Dynamic Stability of Tailing Dams:
## 3 Analysis of the Test Results  
### Theoretical Analysis of the Law of Pore Pressure Evolution of Tailing Silt  
The tailing silt dynamic pore pressure evolution law is affected by various factors. To thoroughly examine the impact of the confining pressure, dynamic stress, and sample density on the evolution of tailing silt's dynamic pore water pressure law, in this study, we adopted limit equilibrium theory to analyze the pore pressure value of the material under limit equilibrium (i.e., when the tailing silt is initially damaged), or, in other words, when the critical pore water pressure, \(\mu_{cr}\), is affected by various factors [21]. The static limit equilibrium of the material is applicable to the dynamic test as well, and the Mohr-Coulomb failure envelope of the dynamic load and static load is the same. As shown in Figure 3, stress circle 1 represents the stress state before vibration and stress circle 2 represents the maximum stress circle in dynamic load application (i.e., the stress circle whose dynamic stress is equal to its amplitude). During the application of dynamic load, the pore pressure in the sample continues to develop and stress circle 2 continues to move to the failure envelope. When the pore pressure reaches the critical value, stress circle 3 is tangential to the failure envelope. According to limit equilibrium theory, the tailing silt reaches the failure state in this condition. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Test Number** & **Density (g/cm\({}^{3}\))** & **CSR** & \
\begin{tabular}{c} **Consolidation** \\ **Confining Pressure** \\ **(kPa)** \\ \end{tabular} &
\begin{tabular}{c} **Termination of** \\ **Cyclic** \\ **Vibration (N\({}_{t}\))** \\ \end{tabular} \\ \hline
1-1 & & & & 50 & 460 \\
1-2 & & & 100 & 190 \\
1-3 & & & 150 & 97 \\
1-4 & & & 200 & 35 \\
1-5 & 1.62 & & 250 & 165 \\
1-6 & & & 300 & 295 \\
1-7 & & & 400 & 510 \\
1-8 & & & 600 & 980 \\
1-9 & & & 1000 & No liquefaction \\ \hline
2-1 & & & & 50 & 670 \\
2-2 & & & 100 & 220 \\
2-3 & & & 150 & 380 \\
2-4 & 1.67 & & 200 & 650 \\
2-5 & & & 300 & 1120 \\
2-6 & & & 600 & No liquefaction \\
3-1 & 1.70 & & 100 & 483 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Experimental scheme. Figure 3: Critical pore water pressure under limit equilibrium. According to the geometric conditions shown in Figure 3, the critical pore water pressure in the limit equilibrium state can be deduced as follows:  
\[u_{cr}=\frac{\sigma_{1}+\sigma_{3}}{2}+\frac{c^{\prime}}{\tan\varphi^{\prime}} -\frac{\sigma_{1}-\sigma_{3}+\sigma_{d}(1-\sin\varphi^{\prime})}{2\sin\varphi^ {\prime}} \tag{1}\]  
In Formula (1), \(c^{\prime}\) and \(\varphi^{\prime}\) are the effective cohesion and effective internal friction angle of the tailing silt sample, respectively; \(\sigma_{d}\) is the dynamic stress amplitude; and \(\sigma_{1}\) and \(\sigma_{3}\) represent the axial compression and confining pressure, respectively. This test is isobaric consolidation (\(K_{c}=1.0\)), and \(\sigma_{1}=\sigma_{3}\); hence, Formula (1) can be simplified as:  
\[u_{\sigma}=\sigma_{3}+\frac{c^{\prime}}{\tan\varphi^{\prime}}+\frac{\sigma_{d }(\sin\varphi^{\prime}-1)}{2\sin\varphi^{\prime}} \tag{2}\]  
According to Formula (2), the critical pore water pressure is linked to the effective shear strength indices, \(c^{\prime}\) and \(\varphi^{\prime}\); the dynamic stress, \(\sigma_{d}\); and the confining pressure, \(\sigma_{3}\), during the test. Therefore, when the influence of a single factor on the critical pore water pressure is analyzed, other factors are assumed to remain unchanged. (1) Confining pressure's effect on critical pore pressure  
Formula (2) can be obtained as follows: Under the condition that other factors remain unchanged, the critical pore pressure, \(\mu_{cr}\), is proportional to the confining pressure, \(\sigma_{3}\). The greater the confining pressure applied to the sample, the greater the critical pore pressure, \(\mu_{cr}\), and the stronger the liquefaction resistance of tailing silt.