## 2 Fundamentals of Non-Equally Proportional Discounting of Strength Parameters
### Reduction in Tensile Strength Parameters  
In computational analysis, there are different views on the degree to which the slope safety factor is affected by tensile strength for different geotechnical properties, heights, and slope angles. For instance, [PERSON] et al. [22] analyzed the impact of tensile strength on the stability of soil slopes with a high liquid limit. They reduced the tensile strength of the soil from 10 kPa to 0.1 kPa or increased the tensile strength of the soil from 10 kPa to 10,000 kPa and observed that the coefficient of safety for slope stability varied by approximately 0.01 in both upward and downward directions. [PERSON] et al. [23] proposed two strength criteria, one considering tensile damage and the other not considering tensile damage. The results showed that the presence or absence of tensile damage has little effect on the factor of safety calculation of homogeneous soil slopes when using the mechanics of the continuum medium with the ideal elasto-plasticity model to simulate the slope failure problem. Therefore, under general static conditions, ignoring the effect of the tensile strength of the slope on stability can still meet engineering needs. However, the tensile damage effect should not be neglected when evaluating slope stability under dynamic loading. [PERSON]. [18] believed that the damage mechanism of a slope under seismic action is manifested in upper tensile damage and lower shear damage to the slope body. The slope mainly experiences tensile damage within a certain depth downward at the top, and the shear slip zone extends upward at the foot of the slope. The two eventually connect to form a penetrating rupture surface, and the author suggested considering the strength reduction method (fully dynamic loading) in slope stability evaluation under dynamic loading. The strength discounting method (the complete dynamic analysis method) is  
\[SRF=\frac{c_{0}}{c_{i}}=\frac{\tan\varphi_{0}}{\tan\varphi_{i}}=\frac{T_{0}}{T_ {i}} \tag{10}\]  
where \(T_{i}\) is the tensile strength after discounting, and \(T_{0}\) is the initial tensile strength parameter. The physical significance of the other parameters is described in the previous section. In the literature [18], the slope's damage mechanism under dynamic conditions is joint damage due to tension and shearing, so it is necessary to consider reducing tensile strength. However, tensile strength is often used to ensure equal proportionality of the reduction in tensile strength, ignoring the intrinsic connection between the reduced tensile strength, cohesion, and the angle of internal friction. Assuming that the geotechnical cohesion of the slope is c, the angle of internal friction is \(\varphi\), the tensile strength is T, and the unit weight is \(\gamma\), the potential slip surface when the slope is in the state of ultimate equilibrium consists of lines AB and BC, in which line AB characterizes the tension crack at the top of the slope, and line BC indicates that it is subjected to shear damage (Figure 3). Figure 3: A schematic diagram of the potential slip surface. Assuming that the depth of point B is \(Z_{0}\), and the principal stress is negative for tensile stress and positive for compressive stress, the principal stress at point B can be expressed by the following equations. Figures 3 and 4 show that point A is located in the tensile damage zone, point C is located in the shear damage zone, and point B is situated in the transition zone between tensile and shear damage.