Shear Strength Applications in Plastic and Limit Equilibrium - Unsaturated Soil Mechanics in Engineering Practice - page 638

The horizontal interslice normal forces E L and E R can-
cel when summed over the entire sliding mass. Substituting
Eq. 12.71 for the mobilized shear force S m into Eq. 12.83
and replacing the σ n β term with N gives

The assumption is made that the interslice shear force X
can be related to the interslice normal force E by a mathe-
matical function (Morgenstern and Price, 1965):

X = λf (x)E (12.90)

[ c β cos α

1
F f

(N tan φ − u a tan φ β

+ where:

f(x) = a functional relationship which describes the

manner in which the magnitude of X/E varies
across the slip surface and

N sin α

u w ) tan φ b β) ]

+ (u a − = A L + (12.84)

where:

λ = a scaling constant which represents the percent-

age of the function f(x) used for solving the
factor-of-safety equations.

F f = factor of safety with respect to force equilibrium.

There are a number of reasonable mathematical functions
that can be assumed to describe the direction of the interslice
forces. However, an unrealistic assumption of the inter-
slice force function can result in convergence difficulties
when solving nonlinear factor-of-safety equations (Ching
and Fredlund, 1983). Morgenstern and Price (1967) sug-
gested that the interslice force function should be related
to the shear and normal stresses on vertical slices through
the soil mass. Maksimovic (1979) used the finite element
method and a nonlinear characterization of the soil to com-
pute stresses in a soil mass. These stresses were then used
in the limit equilibrium slope stability analysis.
A generalized interslice force function f(x) was pro-
posed by Fan et al. (1986). The function was based on
two-dimensional finite element analyses of a linear elastic
continuum. The normal stresses in the x -direction and the
shear stresses in the y -direction were integrated along vertical
planes within a sliding mass in order to obtain normal and
shear forces, respectively. The ratio of the shear force to
the normal force along each vertical section provided the
direction of the resultant interslice forces. Figure 12.74
illustrates a typical interslice force function for one slip
surface through a relatively steep slope.
The analysis of many slopes showed that the interslice
force function could be approximated by an extended form
of an error function equation. Inflection points were close
to the crest and toe of the slope. The slope of the resultant
interslice forces was steepest at the midpoint and tended
toward zero at some distances behind the crest and beyond
the toe of the slope. The mathematical form for the empirical
interslice force function can be written as follows:

Rearranging Eq. 12.84 yields

c β cos α

N

u w β tan φ b
tan φ

+ −

u a β 1

tan φ cos α

tan φ b
tan φ

− −

F f = N sin α (12.85)

A L +

In the case where the pore-air pressure is atmospheric (i.e.,
u a = 0), Eq. 12.85 reverts to the following form:

c β cos α

N tan φ cos α

u w β tan φ b

+ −

tan φ

F f = N sin α

A L +

(12.86)
The φ b value is equal to the φ value when the pore-
water pressure is positive. Equation 12.86 is the same for
both circular and composite slip surfaces.

12.5.10 Interslice Force Function
The interslice normal forces E L and E R are computed from
the summation of horizontal forces on each slice:

E R − E L = N cos α tan α − S m cos α (12.87)

Substituting Eq. 12.73 for the N cos α term in Eq. 12.87
gives the following equation:

E R − E L =

[ W

− (X R − X L ) − S m sin α ]tan α − S m cos α

(12.88)

e − (C n ω n )/ 2

f(x) = (12.91)

Rearranging Eq. 12.88 gives

where:
e

S m
cos α

E R = E L + [ W − (X R − X L ) ]tan α − (12.89)

= base of the natural logarithm, 2.71828,

C = variable to define the inflection points of the func-

tion, The interslice normal forces are calculated by integrating

across the slope. The left interslice normal force on the first
slice is equal to any external water force which may exist,
A L , or it is set to zero when there is no water present in the
tension crack zone.

n = variable to specify the flatness or sharpness of cur-

vature of the function, and

ω = dimensionless x -position relative to the midpoint of

the slope.