Managing risk and achieving reliable geotechnical designs using Eurocode 7 - Risk and Reliability in Geotechnical Engineering - page 405

For a 1D potential slip curve in a 2D problem, it is in general very difficult to determine
Γ 2 . However, with a first-order approximation, the isotropic equivalent δ E can be computed
(El-Ramly et al., 2006) from

(10.13)

δ = δ δ

E y z

so that Γ 2 can be approximated as

(10.14)

Γ S

2

= [ 2 L / δ − 1 + exp( − 2 L / δ )] /( 2 L

2

/) δ

2

v E v E v E

where L v is the length of the potential slip curve.
For a 2D potential slip plane in a 3D problem, for example, a slope failure on a planar
surface, let the plane have extent = L x and SOF = δ x in the direction, x along the slope per-
pendicular to the cross section y-z through the slope, and length = L splane and SOF = δ splane
in the cross-section. Then,

(10.15)

ΓΓΓ

2

=

2 2

splane

S x

where

2 2 2

Γ x = [L/ 2 δ − 1 + exp( − 2 L/ )] (L/) δ 2 δ (10.16)

x x x x x x

(

)

Γ splane

2

= [L 2 / δ − 1 + exp( − 2 L/ δ )] 2 L

2

/ δ

2

(10.17)

splane splane splane splane spla ne splane

δ splane can be determined using equations similar to Equation 10.12 .
For a 2D potential slip (curved) surface in a 3D problem, such as that shown in Figure
10.4 , it is again in general very difficult to derive Γ 2 , but with a first-order approximation,
it can be expressed as

(10.18)

ΓΓΓ

2

=

2 2

S x scurve

where Γ 2 is the variance reduction factor in the direction along the slope and Γ scurv 2 is the vari-
ance reduction factor along the slip curve. Γ 2 can be determined using an equation the same as
Equation 10.16 , where L x is the extent of the failure surface in the x-direction along the slope
perpendicular to the cross-section, as shown in Figure 10.4 , and δ x is the SOF in that direction.
Γ scurve

2

can be determined using an equation similar to Equation 10.14 , where δ = δ δ and

E y z

L scurve is the length of the slip surface in the cross-section plane through the slope.

L x

L v

Figure 10.4 Slope stability example showing the scales of fluctuation and autocorrelation lengths along the
slope perpendicular to the slope cross-section and along the failure surface.