Civil Engineering Reference
In-Depth Information
(a)
(b)
P
n
tan
d
Q
n
∆
x
P
n
P
n
Q
n
-1
M
n
P
n
-1
y
n
n
L
n
W
n
P
n
-1
p
P
n
-1
tan
d
R
n
k
n
R
n
S
n
(c)
Q
n
P
n
W
n
n
R
n
tan
b
T
Q
n
-1
P
n
-1
R
n
L
1
T
Figure 9.9
Limiting equilibrium conditions for toppling and sliding of
n
th block: (a) forces acting on
n
th block;
(b) toppling of
n
th block; (c) sliding of
n
th block (Goodman and Bray, 1976).
respectively,
When the block under consideration is one of the
sliding set (Figure 9.9(c)),
R
n
=
W
n
cos
ψ
p
+
(P
n
−
P
n
−
1
)
tan
φ
d
(9.21)
S
n
=
R
n
tan
φ
p
(9.24)
S
n
=
W
n
sin
ψ
p
+
(P
n
−
P
n
−
1
)
(9.22)
However, the magnitudes of the forces
Q
n
−
1
,
P
n
−
1
and
R
n
applied to the sides and base of the
block, and their points of application
L
n
and
K
n
, are unknown. Although the problem is
indeterminate, the force
P
n
−
1
required to pre-
vent sliding of block
n
can be determined if it is
assumed that
Q
n
−
1
Considering rotational equilibrium, it is found
that the force
P
n
−
1
that is just sufficient to prevent
toppling has the value
P
n
−
1,
t
=[
P
n
(M
n
−
x
tan
φ
d
)
+
(W
n
/
2
)
P
n
−
1
)
. Then the
shear force just sufficient to prevent sliding has
=
(
tan
φ
d
·
(y
n
sin
ψ
p
−
x
cos
ψ
p
)
]
/L
n
(9.23)
