## 2 Safety Factor of a Slope Reinforced by Prestressed Anchor Cables
### Determination of the Safety Factor of the Slope  
At present, there are two main approaches used for determining the safety factor of a slope based on numerical methods (finite element or finite difference). For the inflection point method, the strength parameters of slope rock and soil masses are reduced according to a certain value to calculate the displacement of a slope foot. On this basis, the relationship between the displacement of the slope foot and the strength reduction factor was drawn and then the strength reduction factor corresponding to the inflection point (a point at which displacement suddenly varies) of the curve functions as the safety factor of the slope [31]. A convergent method [32] was also used wherein the strength parameters of slope rock and soil masses were reduced according to a certain value and whether the calculation can be convergent or not (whether unbalanced force is less than \(1.0\times 10^{-5}\) or not is regarded as convergence criterion in FLAC(tm)) is considered as the control standard for slope stability. The corresponding strength reduction factor when the calculation is exactly convergent is considered to be the safety factor for the stability of the slope. In this paper, there is some difference in the calculation process compared to the conventional strength reduction method. Because the slip surface is determined in advance, only the strength parameters of the slip surface need to be reduced and that of the rock mass do not need to be reduced. No matter which method we use to find the safety factor of the slope, the first step is the same in that the initial stress field should be calculated first and the initial deformation should be eliminated before reducing the strength of the slip surface. Then the second step is to apply the prestress anchor cables and reduce the strength of the slip surface to let it slide. Comparing to the relative larger displacement on the slip surface, the deformation of the rock mass caused by the sliding is small enough to be ignored. So the increase of the inner force of the anchor cables can be regarded as only being caused by the sliding of the slip surface. Figure 3 shows the relationship between the displacement of slope foot without being reinforced by prestressed anchor cables and the safety factor of the slope. By utilising the inflection point method, it can be seen that the safety factor (1.0) of the slope corresponded to the inflection point of the curve. When the safety factor was greater than 1.0, the displacement of the slope foot rapidly rose, implying that the slope was at its stability limit without the reinforcement effect of prestressed anchor cables, with a safety factor of 1.0. Figure 4 displays the relationship between the displacement of slope foot reinforced by prestressed anchor cables and the strength reduction factor (calculated by applying the inflection point method). The prestress applying on all cables was 500 kN. As shown in the figure, in the presence of prestressed anchor cables, the safety factor of the slope was greatly increased. The point with the strength reduction factor of 1.8 in the figure corresponded to the inflection point of the curve. On the condition that the strength reduction factor exceeded 1.8, the displacement of slope foot rose rapidly, indicating that the safety factor of the slope reinforced by prestressed anchor cables was determined as being 1.8 by using the inflection point method. Figure 5 presents the relationship between the displacement of slope foot reinforced by prestressed anchor cables and the strength reduction factor (calculated by convergence method). The prestress applying on all cables is 500 kN. It can be seen from the figure that the inflection point of the curve appeared when the strength reduction factor was 1.59. This indicated that the strength reduction factor (1.59) corresponded to the limit convergent state during the calculation by using the convergence method.