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In Bishop's Simplified Method the effect of the inter-slice force is eliminated by assuming that the vertical component of the inter-slice forces is zero. The forces on a typical slice are shown in Figure 3 and consist of:

1. the slice weight, W
   
2. the pseudo-static seismic force, kW, in which k is the seismic coefficient
   
3. the pore pressure force, U ( = u . l )
   
4. the effective normal force on base, P'
   
5. the mobilized shear force,
   
6. the resultant of boundary forces perpendicular and parallel to the top of the slice, N and M
   
7. the horizontal slide forces, E_n and E_n+1
   

Figure 3 Force polygon for Bishop's Simplified Method.

Solving force equilibrium in the vertical direction, thereby eliminating the side forces, yields;

(4)

or rearranging and solving for P':

(5)

Once the normal force at the base of each slice is found, overall moment equilibrium yields and implicit expression for the factor of safety:

(6)

where y_k, a_N and a_M are appropriate moment arms and is given by

(7)

Whitman and Bailey (1967) point out a rare numerical difficulty with this method. When is negative the possibility exists that the denominator in Equation 5 could be negative, or worse, zero. When this occurs, a warning is printed and the resisting moment is set to zero.

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