Stability of Slopes - Elements of Soil Mechanics - page 402

This formula gives a solution generally known as the conventional method which allows rapid determina-
tion of F when sufficient slip circles are available to permit the determination of the most critical. For
analysing the stability of an existing tip it should prove perfectly adequate.

Bishop's routine, or rigorous, method

The formula for the conventional method of analysis can give errors of up to 15% in the value of F obtained,
although the error is on the safe side since it gives a lower value than is the case. In the construction of
new embankments and earth dams, however, this error can lead to unnecessarily high costs, and it
becomes particularly pronounced with a deep slip circle where the variations of α over the slip length
are large.
Returning to the equation:

R
Wx

∑ ′ + −

′

F = ∑ [ c l ( P ul ) tan ] φ

Let the normal effective force, (P − ul) = P ′ .

Resolving forces vertically:

W P = cos α + T sin α

Now

= ′ +

P P ul

and

1 (

′ + ′

T =

F c l P

tan ] φ ′

P ′ tan φ ′

+ ′

c l
F

+ ′ W ul = cos α P cos α + sin α sin α

F

+ ′

′

c l sin α tan φ

= ul cos α + ′ P co s α + sin α

F F

+ ′

′

c
F

t an φ

+ ′

= l ul= cos α sin α P cos α + sin α

F

+ ′

c
F

W l u − cos α sin α

⇒ ′ = P

′

tan φ sin α

cos α +

F

Substituting P ′ for (P − ul) in the original equation:

R
Wx

∑ [

F = ∑ c l ′ + − ( P ul ) tan ] φ ′

− ′

c l
F

′

W ul − cos α sin α tan φ

R
Wx

∑

′ +

F = ∑ c l

′

tan sin φ α

cos α +

F