Civil Engineering Reference
In-Depth Information
Figure 19.8
Vertical stress in the ground.
The variation of vertical total stress with depth in the ground was given in Sec. 6.2.
From Fig. 19.8 the vertical stress on an element at a depth
z
is
σ
=
γ
z
+
q
+
γ
w
z
w
(19.17)
v
where
q
is a uniform surface stress and
z
w
is the depth of water above ground level.
For drained loading the effective vertical stress is given by
σ
v
=
σ
−
u
(19.18)
v
where
u
is the (steady state) pore pressure.
In Fig. 19.9(a) there are two regions A and B separated by a discontinuity represented
by a single bold line; the stresses in each region are uniform and are characterized by
the magnitudes and directions of the major principal stresses
σ
1b
as shown.
The rotation in the direction of the major principal stress across the discontinuity is
δθ
=
θ
b
−
θ
σ
1a
and
a
. The Mohr circles of total stress are shown in Fig. 19.9(b). The point
C represents the normal and shear stresses on the discontinuity and the poles of the
circles are found by drawing
P
a
P
b
parallel to the discontinuity in Fig. 19.9(a).
Hence the directions of the major principal planes are given by the broken lines in
Fig. 19.9(b) and, from the properties of the Mohr circle construction given in Sec. 2.5,
we can mark 2
−
C
−
θ
a
and 2
θ
b
, the angles subtended by
σ
1a
and
σ
1b
, and the normal stress
on the discontinuity.
As usual it is necessary to consider undrained and drained loading separately.
Figure 19.10 shows the analysis for undrained loading. Both Mohr circles of total
stress touch the failure line given by Eq. (19.1). From the geometry of Fig. 19.10(b),
noting that AC
=
s
u
,
δ
s
=
2
s
u
sin
δθ
(19.19)
Hence the change of total stress across a discontinuity is simply related to the rotation
δθ
of the direction of the major principal stress.
Figure 19.11 shows the analysis for drained loading. Both Mohr circles of effective
stress touch the failure line given by Eq. (19.2) and the angle
ρ
defines the ratio
τ
n
/
σ
n
