## 4 Dynamic Analysis of Debris Flows in Niumiangou Valley
### Debris Flow Impact Force  
Debris flow impact force is an important parameter in the design of debris flow prevention projects, and can be mainly divided into the debris flow liquid dynamic pressure and rock impact force.  
1. Liquid dynamic pressure of debris flow. The liquid dynamic pressure of debris flow can be calculated by the following equation [26]: \[\sigma=K_{\rm C}\gamma_{\rm D}V_{\rm C}^{2}\] (6) where \(\sigma\) is the debris flow liquid dynamic pressure, Pa; \(K_{\rm C}\) is the non-uniform coefficient of debris flow fluid and the other symbols have the same meaning as above.  
According to the reference [27; 28], the non-uniform coefficient \(K_{\rm C}=3\) of debris flow fluid is selected, and the liquid dynamic pressure \(\sigma=3.50\times 10^{6}\) Pa of the Niumiangou valley \(8\cdot 14\) debris flow is calculated according to Equation (6).  
Impact force of debris flow boulders.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Dynamic Parameters**} & \multicolumn{4}{c}{**Computation**} \\ \hline Volumetric weight of debris flow/(g/cm\({}^{3}\)) & & & 2.20 & \\ Average velocity of debris flow/(m/s) & & & 23.04 & \\ Liquid dynamic pressure of debris flow/(Pa) & & & \(3.5\times 10^{6}\) & \\ Impact force of debris flow stone/(N) & & & \(1.77\times 10^{8}\) & \\ The maximum height of the debris flow rises/(m) & & & 27.08 & \\ The super elevation in bend of debris flow /(m) & & & 28.07 & \\ \hline Different frequencies/\% & 10 & 5 & 2 &
\begin{tabular}{c} \(1(8\cdot 14\) actual outbreak \\ frequency of debris flow) \\ \end{tabular} & 0.5 \\ \hline Storm flood design flow/(m\({}^{3}\)/s) & 16.42 & 19.04 & 22.27 & 25.16 & 27.95 \\ Debris flow peak discharge/(m\({}^{3}\)/s) & 283.2 & 328.3 & 384.0 & 433.9 & 482.0 \\ Total amount of debris flow/(10\({}^{4}\)m\({}^{3}\)) & 26.91 & 31.21 & 36.50 & 41.24 & 45.81 \\ Total amount of solid materials showed out by a debris flow/(10\({}^{4}\)m\({}^{3}\)) & 19.11 & 22.16 & 25.92 & 29.28 & 32.53 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The results of dynamic parameters of the Niumiangou valley \(8\cdot 14\) debris flow.  
2. The impact force of debris flow boulders is calculated using Equation (7) [29; 30; 31]: \[F=\rho_{\rm d}CV_{\rm d}A_{\rm d}\] (7) where \(F\) is the impact force of boulders in the debris flow, N; \(\rho_{\rm d}\) is the boulder density in the debris flow, kg/m\({}^{3}\); \(C\) is the longitudinal wave velocity, m/s; \(A_{\rm d}\) is the contact area between the boulder and debris flow, calculated according to 10 % of the plane area of the center and short diameter of the largest stone, m\({}^{2}\); \(V_{\rm d}\) is the velocity of the boulder in debris flow, m/s, according to reference [23], which is calculated according to Equation (8); and the meaning of the other symbols is the same as above. \[V_{\rm d}=\alpha\sqrt{d_{\rm max}}\] (8) where \(\alpha\) is the friction coefficient, within \(3.5\leq\alpha\leq 4.5\) and \(d_{\rm max}\) is the largest stone particle size in debris flow, m.  
The field investigation of the Niumiangou valley \(8\cdot 14\) debris flow disaster reveals that the largest boulder in the debris flow was phyllite, the longest diameter was 4.9 m, the volume was 89 m\({}^{3}\) and, according to reference [32], its density and longitudinal wave velocity can be selected as \(\rho_{\rm d}=2.75\times 10^{3}\) kg/m\({}^{3}\) and \(C=4000\) m/s; the friction coefficient \(\alpha\) is the average value of \(\alpha\), that is, \(\alpha=4\). According to Equation (8), the velocity of the stone is calculated as \(V_{\rm d}=8.85\) m/s; according to the longest diameter and volume of the largest boulder, the contact area \(A_{\rm d}=1.82\) m\({}^{2}\) was calculated. In summary, the maximum impact force of boulders in the Niumiangou valley \(8\cdot 14\) debris flow is calculated according to Equation (7) as \(F=1.77\times 10^{8}\)N.