METHOD OF SLICES
Bishop’s Method:
Bishop (1955) assumed a circular slip surface, as shown in Figure 16.5a. Let us apply the equilibrium equations to the forces on the slice shown in Figure 16.5c, assuming that Ej and Ej11 and Uj and Uj11 are collinear, Nj acts at the center of the arc length, that is, at lj/2, and (Js)j = 0.
Summing forces vertically, we get
The force due to the porewater pressure (Uj) is Uj = ujlj. From the principle of effective stress,
Combining Equations , we get
For convenience, let us defi ne the force due to the porewater as a function of Wj as
where ru is called the porewater pressure ratio. Substituting Equation yields
Bishop considered only moment equilibrium, such that, from Figure
where xj is the horizontal distance from the center of the slice to the center of the arc of radius R, and Tj is the mobilized shear force. Solving for Tj from Equation and noting that xj =R sin uj, we get
Recall from Equation that the factor of safety is defined as
where Tf is the soil shear force at failure. In developing the governing equation for FS, we will first consider effective stress, and later, total stress. For an ESA,
Substituting Equation yields
Putting
we can write N'j as
Substituting Equation gives
Combining Equations yields
where (su)j is the undrained shear strength of the soil along the slip surface within the slice. Combining Equations yields
Failure
surface proposed by
Janbu and forces on a
slice of soil.