Environmental Engineering Reference
In-Depth Information
or for the case of
φ
0,
H
cr
2.67
c
/
γ
t
(9.15)
Field observations indicate that the tension crack depth
z
c
ranges from 1/3
H
to 1/2
H
. In
practice,
z
c
is often taken as 1/2
H
cr
of an unsupported vertical cut or as
z
c
2
c
/
γ
(9.16)
which is considered conservative (Tschebotarioff, 1973).
Case 7
: Multiple planar failure surfaces are illustrated as follows, relationships not
included:
Active and passive wedge force system applicable to rock or soil and rock slopes
●
General wedge or sliding block method applicable to soil formations and earth
dams —
Figure 9.73
●
Intersecting joints along a common vertical plane —
Figure 9.74
●
Triangular wedge failure, applicable to rock slopes —
Figure 9.75
●
Finite Slope: Cylindrical Failure Surface
In rotational slide failures, methods are available to analyze a circular or log-spiral failure
surface, or a surface of any general shape. In all cases, the location of the critical failure
surface is found by trial and error, by determining the factor of safety (FS) for various trial
positions of the failure surface until the lowest value of the FS is reached. The forces act-
ing on a free body taken from a slope are given in
Figure 9.76.
Most modern analytical methods are based on dividing the potential failure mass into
slices, as shown in
Figure 9.77.
The various methods differ slightly based on the force sys-
tem assumed about each slice
(Figure 9.78)
.
Iterations of the complex equations to find the
“critical circle” has led to the development of many computer programs for use with the
Personal Computer. Equation 9.17 given under the Janbu method is similar to the
Simplified Bishop. In most cases, for analysis, the parameters selected for input are far
more important than the method employed. The most significant parameter affecting the
FS is usually the value input for shear strength; assuming even a small amount of cohe-
sion can result in FS
1.2 rather than 1.02, if only internal friction is assumed.
Ordinary method of slices
: In 1936, Fellenius published a method of slices based on cylin-
drical failure surfaces which was known as the Swedish Circle or Fellenius method.
Modified for effective stress analysis, it is now known as the Ordinary Method of Slices.
As illustrated in Figure 9.77, the mass above a potential failure surface is drawn to scale
and divided into a number of slices with each slice having a normal force resulting from
its weight. A flow net is drawn on the slope section (Figure 9.77a), or more simply, a
phreatic surface is drawn. Pore pressures are determined as shown in Figure 9.77b. The
equilibrium of each slice is determined and FS found by summing the resisting forces and
dividing by the driving forces as shown in Figure 9.77c. The operation is repeated for other
circles until the lowest safety factor is found. The method does not consider all of the
forces acting on a slice (Figure 9.78a), as it omits the shear and normal stresses and pore-
water pressures acting on the sides of the slice, but usually (although not always) it yields
conservative results. However, the conservatism may be high.
Bishop's method of slices:
This method considers the complete force system (Figure 9.78b),
but is complex and requires a computer for solution. The results, however, are substantially
