Rigorous solution of slope stability under seismic action

Oblique and eccentric bearing capacity of embedded strip footings using the method of rigorous characteristics

“…Geometry equation 19–21 $$\begin{equation}{\lambda _x} = \frac{{\partial {v_x}}}{{\partial x}},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\lambda _y} = \frac{{\partial {v_y}}}{{\partial y}}{\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\varpi _{xy}} = (\frac{{\partial {v_x}}}{{\partial y}} + \frac{{\partial {v_y}}}{{\partial x}})\end{equation}$$…”

“… Geometry equation 19–21 $$\begin{equation}{\lambda _x} = \frac{{\partial {v_x}}}{{\partial x}},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\lambda _y} = \frac{{\partial {v_y}}}{{\partial y}}{\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\varpi _{xy}} = (\frac{{\partial {v_x}}}{{\partial y}} + \frac{{\partial {v_y}}}{{\partial x}})\end{equation}$$ where $${\lambda _x}$$ and $${\lambda _y}$$ are the linear strain rates along the x‐axis and y‐axis directions, respectively; $${\varpi _{xy}}$$ is the shear strain rate in the x‐y plane; $${v_x}$$ and $${v_y}$$ are the basic velocity components along the x‐axis and y‐axis directions, respectively. Plastic flow rule 19–21 $$\begin{equat...$$…”

“…Shen 34 demonstrated that normal and shear stresses on both sides of the stress discontinuity line should be consistent, but the tangential stresses are discontinuous on both sides. Li et al 19 . proposed the relationship of equivalent stresses $${\sigma _e}$$ on both sides of the stress discontinuity line, which can be established as $$\begin{equation}\sigma _e^ + = {\left( {\sqrt {1 + \frac{{{{\cos }^2}{\varphi ^*}}}{{{{\tan }^2}({\theta ^ + } - {\theta ^ - })}}} + \sin {\varphi ^*}} \right)^2}\frac{{{{\sin }^2}({\theta ^ + } - {\theta ^ - })}}{{{{\cos }^2}{\varphi ^*}}}\sigma _e^ - \end{equation}$$where $$\sigma _e^ - $$ and $${\theta ^ - }$$ are the equivalent stress and the direction angle in the zone OEA, while $$\sigma _e^ + $$ and $${\theta ^ + }$$ are the corresponding variables in the zone OBA.…”

“…In order to achieve the distribution of the stress discontinuity line, it is usually assumed to be tangent to the initial boundary 19,34,35 . In this study, the stress discontinuity line OA is assumed to be tangent to the ground at point A.…”

“…In this study, the stress discontinuity line OA is assumed to be tangent to the ground at point A. Then, some scholars 19,34,35 indicated that the distribution of the stress discontinuity line in the slip line field can be obtained by Equations (13) and (16). As shown in Figure 6, when the effect of embedded soils is assumed to be a surcharge loading $${q_d} = \gamma d$$ ( d is the embedded depth) without considering the soil shear strength, the initial boundary AOD will be continuous, and the slip line field in the passive zone AOB can be directly established without the stress discontinuity line.…”