Environmental Engineering Reference
In-Depth Information
Table 2.3
Measured shear strength parameters for silty clay in Taiyuan, China
Q (15 data)
CQ (15 data)
UU (15 data)
CU (15 data)
Test
number
c
(kPa)
ϕ
(°)
c
(kPa)
ϕ
(°)
c
(kPa)
ϕ
(°)
c
(kPa)
ϕ
(°)
1
32.2
8.2
35.1
6.2
25.3
16.0
40.1
7.1
2
40.2
10.5
60.3
16.3
58.4
6.5
60.2
8.2
3
30.1
15.2
32.6
15.2
43.5
9.6
60.4
18.1
4
35.6
12.8
55.6
22.6
52.1
10.7
70.6
11.5
5
49.3
8.1
75.2
9.1
60.0
6.5
80.5
8.3
6
23.5
13.6
29.8
15.4
40.3
6.9
45.1
10.1
7
17.6
24.5
25.7
35.6
25.7
16.0
45.3
20.6
8
40.1
15.1
45.7
25.1
35.6
7.5
36.3
15.7
9
19.8
16.8
22.6
16.2
20.2
12.1
28.3
19.3
10
20.6
18.1
35.3
18.2
33.3
9.2
53.2
19.5
11
16.3
20.3
30.4
30.8
21.6
15.8
16.1
26.4
12
23.7
23.1
41.5
31.7
24.1
14.9
30.4
22.5
13
19.1
26.2
31.2
25.4
49.6
15.7
63.1
25.1
14
30.5
16.2
51.2
23.5
49.1
13.5
53.5
19.1
15
10.2
25.0
19.5
30.4
28.7
15.8
30.1
23.7
COV
0.40
0.35
0.39
0.40
0.36
0.33
0.37
0.38
−
0.813
−
0.367
−
0.591
−
0.504
ρ
−
0.676
−
0.257
−
0.425
−
0.333
τ
Source: After Li XY et al.
Chinese Journal of Geotechnical Engineering
2000; 22(6): 668-672 (in Chinese).
Note: Q, Quick direct shear test; CQ, consolidated-quick direct shear test; UU, unconsolidated-undrained triaxial compres-
sion test; CU, consolidated-undrained triaxial compression test.
2. Estimate the distribution parameters
p
and
q
underlying the four candidate distribu-
3. Evaluate the AIC and BIC values associated with the four candidate distributions using
Equations 2.25
and
2.26
,
respectively.
4. The marginal distribution resulting in the smallest AIC and BIC values is identi-
fied to be the best-fit marginal distribution to the measured data of shear strength
parameters.
As an example, consider the cohesion (
c
) data of CU dataset, the sample mean and sample
standard deviation are computed as μ = 66.483 and σ = 28.778, respectively. The distribu-
tion parameters underlying the Lognormal distribution are then obtained as
p
= 4.111 and
q
= 0.414. Substituting the observed data {
c
i
,
i
= 1, 2, …,
N
} into
f
(
x
;
p
,
q
) of the Lognormal
distribution leads to the logarithm of the likelihood function equal to −337.146. Finally, the
AIC and BIC for the Lognormal distribution are obtained as AIC = −2 × (−337.14 6) + 2 × 2 =
678.292 and BIC = −2 × (−337.14 6) + 2 × ln(64) = 682.611. Following the above procedure,
Table 2.6
shows the AIC and BIC values associated with the four marginal distributions for
various datasets of (
c
, ϕ). Note that both the AIC and BIC values indicate that the Weibull
and Lognormal distributions are the best-fit distributions underlying the CD and UU data-
sets for
c
and ϕ, respectively. For the CU dataset, the best-fit distributions for both
c
and ϕ
are the TruncNormal distributions. To further examine the capabilities of the identified mar-
ginal distributions to fit the measured data,
Figure 2.3
shows the PDFs of the four candidate
marginal distributions along with the histograms of the measured data. It is evident that the
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