Civil Engineering Reference
In-Depth Information
12 5
0 005 31 16
.
From the charts N
=
0 005
.
⇒
F
c
=
=
5 04
.
′
.
× ×
Try F
=
2.0:
tan
φ
0 364
2 0
.
′
=
=
=
0 182
.
(
φ
10 25
.
°
)
F
.
12 5
7 95
.
From the charts N
=
0 0
.
16
⇒
F
c
=
=
1 57
.
′
.
Try F
=
1.9:
tan
φ
0 364
1 9
.
=
=
0 192
.
(
φ
′ = °
11
)
F
.
12 5
6 45
.
From the charts N
0 013
.
F
1 94
.
(
acceptable
)
=
⇒
=
=
′
c
.
Factor of safety for slope
=
1.9
13.3.2 Bishop & Morgenstern charts: homogeneous slope with a constant pore
pressure ratio
If on a trial slip circle the value of F is determined for various values of r
u
and the results plotted, a linear
relationship is found between F and r
u
(see Example
13.5)
. The usual values of r
u
encountered in practice
range from 0.0 to 0.7 and it has been established that this linear relationship between F and r
u
applies
over this range. The factor of safety, F, may therefore be determined from the expression:
F m nr
u
= −
in which m is the factor of safety with respect to total stresses (i.e. when no pore pressures are assumed)
and n is the coefficient which represents the effect of the pore pressures on the factor of safety. These
terms m and n are known as stability coefficients and were evolved by Bishop and Morgenstern (
1960
);
they depend upon c
′
/
γ
H (the stability number), cot
β
(the cotangent of the slope angle, e.g. a 5:1 slope
means 5 horizontal to 1 vertical), and
φ
′
(the effective angle of shearing resistance).
Bishop and Morgenstern prepared charts of m and n for three sets of c
′
/
γ
H values (0.0, 0.025, 0.05),
which are reproduced at the end of this chapter and cover slopes from 2:1 (26.5°) to 5:1 (11.5°). Extrapo-
lation, within reason, is possible for a case outside this range.
Graphs to cover depth factor values, D, up to 1.5 were produced for c
′
/
γ
H
=
0.05, but for the other
two cases D values greater than 1.25 were not calculated as such values are not critical in these instances.
As in Taylor's analysis, the effect of tension cracks has not been included. O'Connor and Mitchell (
1977
)
extended the work of Bishop and Morgenstern to include c
′
/
γ
H
=
0.075 to 0.150.
Determination of an average value for r
u
Generally r
u
will not be constant over the cross-section of an embankment and the following procedure
can be used to determine an average value.
In Fig.
13.26
the stability of the downstream slope is to be determined. From the centre line of the
cross-section, divide the base of the dam into a suitable number of vertical slices (a, b, c, d), and on the
centre line of each slice determine r
u
values for a series of points as shown. Then the average pore pres-
sure ratio on the centre line of a particular slice is
h r
+
h r
+
h r
+
1 1
u
2 2
u
3 3
u
r
=
u
h
+ + +
h
h
1
2
3
