Environmental Engineering Reference
In-Depth Information
V
t
l
Figure 5.3
Footing design.
The LSF describing failure of this foundation is
Q
V
l
g
()
X
=−
2
.
(5.8)
u
where
Q
u
is the ultimate bearing capacity and
V
is the applied vertical loading, which includes
the weight of the foundation. For simplicity, we assume a deterministic
V
= 3000 kN. With
cohesion
c
= 0, the ultimate bearing capacity is
QtNs
=
γ
+
γ
lN is,
γγ
,
(5.9)
u
q
q
where γ = 19.8 kN/m
3
is the unit weight of the soil, and the coefficients are
N
q
= exp(π + tanφ)
tan
2
(45° + (φ/2)),
N
γ
= (
N
q
+ 1)tanφ, and shape factors
s
q
= 1 + sinφ,
s
γ
= 0.7.
The friction angle φ is a spatially variable parameter. Exact computations should explic-
itly address this spatial variability through random fields, as discussed in Section 5.3.1.
However, if we assume highly fluctuating soil properties, then as a first approximation, one
can take the mean value of the friction angle as representative (the approximation is exact
for an uncorrelated soil, i.e., for a correlation length zero; (Griffiths and Fenton 2001)). We
make this assumption here, that is, we replace φ in the above expressions by μ
φ
. However,
it must be reminded that this is an unconservative approximation, and one should make
adjustments when using this approximation in real problems (Griffiths and Fenton 2001).
Owing to the fact that the only random variable in this problem is μ
φ
, a simple solution
for the probability of failure Pr(
F
) = Pr(
g
(
X
) ≤ 0) is available: First, we find the value of
the friction angle for which g = 0 as φ
F
= 21.36°. Failure occurs if the mean friction angle
μ
φ
takes a value smaller than φ
F
, that is, an equivalent expression for the failure event is
F
=
{
µϕ
ϕ
≤
}.
Hence, the probability of failure a priori is
F
Pr()
F
=
Pr(( )
g
X
≤
0
)
=
Pr(
µϕ ϕ
ϕ
≤
)
=
F
′ =×
−
()
2 310
.
3
.
(5.10)
F
µ
ϕ
F
F
µ
ϕ
is the prior cumulative distribution function (CDF); here, the normal CDF
with mean
′ =°
′
where
µ
µ
28
and standard deviation
σ
µ
′ =
234
.
°
.
The corresponding reliabil-
ity index is ′ =−
βΦ
1
−
(.
23
×
10
−
3
)
=
284
.
, with Φ
−1
being the inverse normal CDF. Note
F
µ
ϕ
is a normal CDF, the reliability index can also be computed directly as
β′ = (28° − φ
F
)/2.34° = 2.84.
Since the posterior distribution of μ
φ
conditional on the measurements is the nor-
mal distribution with
′′ =
that, because
′
σ
µ
19. ,
the posterior reliability index is
β″ = (26.08° − φ
F
)/1.39° = 3.40 and the corresponding posterior probability of failure given
the measurements is
µ
µ
26 08
.
°
′′ =
°
and
Pr(
FZ
|
)
=
F
µ
ϕ
ϕ
′′ =×
−
()
34
.
10
4
.
(5.11)
F
Note that the measurements of Illustration 1 resulted in friction angles that are lower
than the prior mean and hence the posterior mean (26.08°) is below the prior mean (28°).
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