# Analysis of the Dynamic Stability of Tailing Dams:
## 4 Hysteresis Curve Analysis for Tailing Silt under Cyclic Loading
### Energy Dissipation Analysis of Typical Tailing Silt  
As shown in Figure 9, the overall development of the hysteresis curves of tailing silt was relatively regular, and the figure is closed. The area enclosed by the hysteresis curves reflects the energy dissipated by the sample under a dynamic cycle. Under dynamic load, the energy dissipated by the sample was closely related to the strain failure of the sample and the liquefaction mechanism of the sample. Therefore, starting with the dynamic stress-dynamic strain relationship, we analyzed the energy dissipation characteristics of tailing silt under different conditions by calculating the area of the hysteretic curves, and the failure mechanism of the tailing silt under cyclic loads was also studied. Figure 9: Hysteresis curves of typical tailing silt at different stages: (**a**) elastic strain stage, (**b**) plastic failure stage, and (**c**) softening stable stage. The calculation of the energy dissipation area primarily adopts the principle of spatial analytic geometry. The area of the triangle sandwiched by vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) is \(1/2\) of the vector modulus of their cross-product, as shown in Formula (4). When the two-dimensional area is calculated, the first two values are 0. \[\overrightarrow{a}\times\overrightarrow{b}= \begin{vmatrix}a_{y}&a_{z}\\ b_{y}&b_{z}\end{vmatrix} i- \begin{vmatrix}a_{x}&a_{z}\\ b_{x}&b_{z}\end{vmatrix} j+ \begin{vmatrix}a_{x}&a_{y}\\ b_{x}&b_{y}\end{vmatrix} k \tag{4}\]  
In this experiment, 20 sets of dynamic stress-strain points were collected per vibration and corresponding polygons were formed. When the load frequency is 1 Hz, the independence of the area calculation of each vibration can be guaranteed. The area of dynamic stress-strain hysteresis curves can be calculated by calculating the polygon area. The calculation of the area of triangle OAB in Figure 10 is shown in Formula (5). \[S_{OAB}=\frac{1}{2}|OA\times OB|=-\frac{1}{2} \begin{vmatrix}\epsilon_{Ad}& \sigma_{Ad}\\ \epsilon_{Bd}&\sigma_{Bd}\end{vmatrix}  \tag{5}\]  
Here, \(\epsilon_{Ad}\) represents the dynamic strain corresponding to point A, \(\sigma_{Ad}\) represents the dynamic stress corresponding to point A, \(\epsilon_{Bd}\) represents the dynamic strain corresponding to point B, and \(\sigma_{Bd}\) represents the dynamic stress corresponding to point B. Since the data collected in this experiment were clockwise, Formula (5) should be multiplied by \(-1/2\). According to the method of superposition of data points and 20 datasets of dynamic stress-strain points, data points were collected for each vibration frequency; the area of the dynamic stress-strain hysteresis curve at vibration number i is shown in Formula (6). \[Si=-\frac{1}{2} \begin{pmatrix}\begin{vmatrix}\epsilon_{1d}&\sigma_{1d}\\ \epsilon_{2d}&\sigma_{2d}\end{vmatrix}+\begin{vmatrix}\epsilon_{2d}&\sigma_{2d }\\ \epsilon_{3d}&\sigma_{3d}\end{vmatrix}+\cdots+\begin{vmatrix}\epsilon_{19d}& \sigma_{19d}\\ \epsilon_{20d}&\sigma_{20d}\end{vmatrix}+\begin{vmatrix}\epsilon_{20d}&\sigma_ {20d}\\ \epsilon_{1d}&\sigma_{1d}\end{vmatrix}\end{pmatrix}  \tag{6}\]  
As is evident from the derivation process, the coordinate quadrant and coordinate origin of the data point in Formula (6) are independent of the relative position of the hysteresis curve; thus, the stress-strain data can be used in the calculation of the energy dissipated by a certain cyclic vibration. Figure 11 shows the dissipated energy, Si, per cycle at \(\sigma_{0}^{\prime}\) = 200 kPa, 400 kPa, and 600 kPa. As the number of cyclic vibrations grew, the energy dissipated with each vibration of the tailing silt gradually increased and tended to stabilize. Based on the size and trend of energy dissipation, the process can be divided into three stages: AB, BC, and CD. In Figure 11, section AB corresponds to the elastic strain stage at the early stage of cyclic loading. At this stage, the hysteresis curves gradually shifted and diffused based on the initial shape, and the maximum positive strain increased marginally.