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General Limit Equilibrium method 1977
Fredlund and Krahn (1977) developed a simpler derivation of Morgenstern and Price’s
method, known as the General Limit Equilibrium method (GLE). The GLE method can use
different interslice force functions such as constant, half-sine, or trapezoidal (GEO-SLOPE
International Ltd, 2008). Two factors of safety equations exist: one for moment equilibrium
the other for force equilibrium.
The normal force is derived from the vertical equilibrium of force either in terms of the
interslice force (E) or the interslice shear force (X):
𝑐 ′ 𝐿 sin𝛼 𝑢 𝐿 sin𝛼 tan𝜙 ′
sin𝛼 tan𝜙
cos𝛼 +
𝑊 − (𝐸𝑖+1 − 𝐸𝑖 ) tan 𝜃 −
𝑐 ′ 𝐿 sin𝛼 𝑢 𝐿 sin𝛼 tan𝜙 ′
sin𝛼 tan𝜙 ′
cos𝛼 +
𝑊 − (𝑋𝑖+1 − 𝑋𝑖 ) −
Factor of safety for moment equilibrium for circular slip circle:
𝐹𝑚 =
∑(𝑐 ′ 𝐿 + (𝑁 − 𝑢 𝐿) tan𝜑 ′ )
∑ 𝑊 sin𝛼
Factor of safety for force equilibrium:
∑(𝑐 ′ 𝐿 + (𝑁 − 𝑢 𝐿) tan𝜙 ′ ) cos𝛼
𝐹𝑓 =
∑ 𝑁 sin𝛼
An iterative process is then used to calculate the value of scaling factor (𝜆) when the factor of
safety for force equilibrium is equal to the factor of safety for moment equilibrium. The first
iteration assumes the vertical shear forces (Xi and Xi+1) are zero, so the horizontal interslice
forces (Ei and Ei+1) can be calculated. The vertical shear forces are then calculated using the
assumed values of the scaling factor (𝜆) and function (𝑓(𝑥)) in Equation 4.30, which then
allows the normal force (N) and subsequently the factors of safety to be calculated
(Albataineh, 2006). The SLOPE/W Engineering Book (2008) states that the GLE method is
“very useful for understanding what is happening behind the scenes and understanding the
reasons for differences between various methods”, however it is not necessarily a method for
routine analysis in practice, due to its inherent costs, time taken, and difficulty in subsequent
interpretation (GEO-SLOPE International Ltd, 2008).