## BISHOP'S METHOD OF SLICES

### Method :

The components V n V_n , V n + 1 V_{n+1} , H n H_{n} , H n + i H_{n+i} contribute to the forces acting on slice (i) .

- Vertical Force Equilibrium:

W i + Δ V i − ( N i ' + u i ⋅ l i ) cos α i − ( C i ' F s l i + N i ' tan φ i ' F s ) sin α i = 0 alignc {W_i + %DELTA V_i - ( N_i^' +u_i cdot l_i ) cos %alpha_i- ( C_i^' over F_s l_i + N_i^' {tan%varphi_i^'} over F_s )sin %alpha_i=0}

N i ' ( cos α i + tan φ i ' sin α i F s ) = W i + Δ V i − u i l i cos α i − C i ' l i F s sin α i N_i^' ( cos %alpha_i + {tan %varphi_i^' sin %alpha_i }over F_s )= W_i+ %DELTA V_i -u_i l_i cos %alpha_i - {C_i^' l_i}over F_s sin%alpha_i

N i ' = W i + Δ V i − u i l i cos α i − C i ' l i F s sin α i cos α i + tan φ i ' ⋅ sin α i F s N_i^'= { W_i+%DELTA V_i-u_i l_i cos %alpha_i - {C_i^'l_i} over F_s sin%alpha_i }over {cos %alpha_i +{ {tan %varphi_i^' cdot sin %alpha_i }over F_s}}

- Moment Equilibrium:

Driving moment:

∑ 1 n W i R sin α i sum from{1} to{n} W_i R sin %alpha_i

Resisting moment (opposing movement):

∑ 1 n R ( C i ' l i F s + N i ' tan φ i ' F s ) sum from{1} to{n} R( {C_i^' l_i}over F_s+ N_i^' {tan %varphi_i^'} over F_s )

By equating moments, substituting (Nᵢ'), and simplifying by R:

F s ( ∑ 1 n W i sin α i ) = ∑ 1 n C i ' l i + W i + Δ V i − u i l i cos α i − C i ' l i F s sin α i cos α i + tan α i ' F s F_s ( sum from{1} to{n} W_i sin %alpha_i )= sum from{1} to{n} C_i^' l_i +{ { W_i +%DELTA V_i-u_i l_i cos %alpha_i - {C_i^' l_i} over F_s sin %alpha_i}}over { cos %alpha_i+ {{tan %alpha_i^'} } over F_s}

The numerator terms can be rewritten with common denominator:

C iˇ ' l i cos α + 1 F s C i ' l i tan φ i ' sin α i + ( W i + Δ V i − u i l i cos α i − 1 F s C i ' l i sin α i ) tan φ i ' C_iˇ' l_i cos %alpha+ 1 over F_s C_i^' l_i tan %varphi_i^'sin %alpha_i +( W_i+%DELTA V_i - u_i l_i cos %alpha_i- {1 over F_s} C_i^' l_i sin %alpha_i)tan %varphi_i^'

Safety factor formula is given by:

F s = 1 ∑ 1 n sin α i ∑ C i ' l i cos α i + ( W i + Δ V i − u i l i cos α i ) tan φ i ' cos α i + 1 F s tan φ i ' sin α i F_s= 1 over { sum from{1} to{n} sin %alpha_i } sum {{C_i^' l_i cos %alpha_i + (W_i+%DELTA V_i - u_i l_i cos %alpha_i) tan %varphi_i^'}over {cos %alpha_i + 1 over F_s tan %varphi_i^' sin %alpha_i} }

This is called the exact Bishop formula.

Solution Procedure:

- Requires iterative calculations since Fs[1] appears on both sides

- Requires additional assumptions to define Vᵢ

Simplified Bishop Method

Assuming Δ V i = V i − V i + 1 = 0 %DELTA V_i=V_i-V_{i+1}=0 , the equation becomes:

F s = 1 ∑ 1 n sin α i ∑ C i ' l i cos α i + ( W i − u i l i cos α i ) tan φ i ' cos α i + 1 F s tan φ i ' sin α i F_s= 1 over { sum from{1} to{n} sin %alpha_i } sum {{C_i^' l_i cos %alpha_i + (W_i - u_i l_i cos %alpha_i) tan %varphi_i^'}over {cos %alpha_i + 1 over F_s tan %varphi_i^' sin %alpha_i} }

Or F s = 1 ∑ 1 n W i sin α i ∑ 1 n C i ' b i + ( W i − u i b i ) tan φ i ' cos α i + 1 F s tan φ i ' sin α i F_s = 1 over { sum from{1} to{n} W_i sin %alpha_i } {sum from{1} to{n} {{ C_i^' b_i +(W_i-u_i b_i)tan%varphi_i^' }over { cos %alpha_i+ 1 over F_s tan %varphi_i^' sin %alpha_i } }}

The expression for F s F_s is not explicit. Therefore, F s F_s cannot be calculated directly. An implicit method will be used in the form F m + 1 = f ( F m ) F_{ m+1 }=f( F_m ) .

The initial value of F s F_s can be taken as F s 0 F_s0 , the value obtained using the Fellenius method. Convergence is generally quite fast. The process is stopped when ( F m + 1 − F m ) (F_ {m+1 }- F_m ) is less than a pre-defined threshold.

### Note :

The Bishop[2] method is more accurate than the Fellenius method, but it requires three to four times more computation (three iterations); the safety factors obtained are generally slightly higher.

### Reminder :

Most often, to avoid excessively increasing the amount of calculation, the most critical slip circle is first determined using the Fellenius method, and then it is verified that the safety factor calculated using the Bishop method is greater than that calculated using the Fellenius method.

If this is not the case, the search for the critical circle must be redone using the Bishop method. (Philipponnat G. & Hubert B, 2000)[3]

## Notes

1. Fs: Factor of Safety

2. 2

   Bishop, A.W. (1955) The Use of the Slip Circle in the Stability Analysis of Slope. Geotechnique, 10, 129-150. - References - Scientific Research Publishing.

3. -

   Philipponnat, G. Hubert B. Fondations et ouvrages en terre, Ed. Eyrolles ... forme ».Guide technique, France, 2000.