Civil Engineering Reference
In-Depth Information
Forces due to self-weight {G}, water load {W} and external forces {P} acting on the
wedge can be combined to a resultant {R}:
{R} = {G} + {W} + {P}.
(11.59)
To check stability, fi rst it has to be investigated whether sliding parallel to the intersec-
tion line of both discontinuities or sliding parallel to one of the two discontinuities is
relevant. To this end, the resultant {R} is resolved into vectors {N
s
} and {T
s
} directed
normal and parallel to the intersection line (Fig. 11.20, above):
{R} = {N
s
} + {T
s
},
(11.60)
in which
{T
s
} = ({R}
⋅
{s})
⋅
{s} = T
s
⋅
{s},
(11.61)
where T
s
is the absolute value of {T
s
} and {N
s
} can also be calculated as a function of
{R} and {s} by inserting (11.61) into (11.60) and solving for {N
s
}.
In the next step, {N
s
} is further resolved into vectors {N
1
} and {N
2
} normal to the
discontinuities D1 and D2 (Fig. 11.20, below):
(11.62)
where N
1
and N
2
are the absolute values of {N
1
} and {N
2
}.
In consideration of (11.59), T
s
, N
1
and N
2
can be expressed in terms of the contributions
of {G}, {W} and {P} parallel to the intersection line of the discontinuities (G
T
and P
T
)
and normal to the discontinuities (G
N1
, G
N2
, P
N1
, P
N2
, W
1
and W
2
), respectively:
T
s
= G
T
+ P
T
,
(11.63)
N
1
= G
N1
+ P
N1
- W
1
,
(11.64)
N
2
= G
N2
+ P
N2
- W
2
.
(11.65)
If {N
1
} and {N
2
} are compressive forces the stability against sliding parallel to the in-
tersection line has to be investigated. In this case it is assumed that the resisting forces
due to the shear strength of the discontinuities acting in the direction opposite to {T
s
}
are fully mobilized. According to the partial safety factor method, stability is assessed
calculating the utilization factor as
(11.66)
in which
φ
D2,d
and c
D2,d
are the design values of the shear parameters of
both discontinuities, and S
D1
and S
D2
are the surfaces of D1 and D2 on which sliding
can take place.
φ
D1,d
, c
D1,d
,
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