Polynomial chaos expansions and stochastic finite-element methods - Risk and Reliability in Geotechnical Engineering - page 287

Table 6.6 Example #2: Two-layer soil layer mass—parameters of the model
Parameter

Notation Type of PDF Mean Value Coefficient of Variation

Upper layer soil thickness t 1 Deterministic 8 m -

Lower layer soil thickness t 2 Deterministic 22 m -

Upper layer Young's modulus E 1 Lognormal 50 MPa 20%

Lower layer Young's modulus E 2 Lognormal 100 MPa 20%

Upper layer Poisson ratio Uniform 0.3 15%

ν 1

Lower layer Poisson ratio Uniform 0.3 15%

ν 2

Load #1 P 1 Gamma 0.2 MPa 20%

Load #2 P 2 Weibull 0.4 MPa 20%

Table 6.7 Settlement of a foundation on a two-layer soil mass—reliability results
Threshold

FORM SORM Importance Sampling

u adm (cm) P f ,FORM P f ,SORM P f , IS β β β

12 1.02 1.09 1.07

1.54 ⋅ 10 − 1 1.38 ⋅ 10 − 1 1.42 ⋅ 10 − 1 [CoV = 1.27%]

15 1.64 ⋅ 10 − 2 2.13 1.36 ⋅ 10 − 2 2.21 1.43 ⋅ 10 − 2 [CoV = 1.67%] 2.19

20 1.54 ⋅ 10 − 4 3.61 1.17 ⋅ 10 − 4 3.68 1.23 ⋅ 10 − 4 [CoV = 2.23%] 3.67

21 3.86 3.94 3.91

5.57 ⋅ 10 − 5 4.14 ⋅ 10 − 5 4.53 ⋅ 10 − 5 [CoV = 2.27%]

The serviceability of this foundation on a layered soil mass vis-à-vis an admissible settle-
ment is studied. The limit state function is defined by

g ( X ) = u adm − u A = u adm − M FE ( E 1 , E 2 , ν 1 , ν 2 , P 1 , P 2 ), (6.82)

in which the admissible settlement is chosen to be 12, 15, 20, and 21 cm. First, FORM and
SORM are applied, together with IS at the design point using n MCS = 10 4 samples. The latter
is considered as the reference solution. Results are reported in Table 6.7 .
Then PC expansions of the maximal settlement M PC are computed using different EDs
of increasing size, namely n = 100, 200, 500, and 1000 using the UQLab platform (Marelli
and Sudret, 2014). The obtained expansion is then substituted for in the limit state function
( Equation 6.82 ) and the associated reliability problem is solved using crude MCS ( n MCS = 10 6
samples). The results are reported in Table 6.8 .

Table 6.8 Settlement of a foundation on a two-layer soil mass—reliability results from PC expansion
(MCS with n MCS = 10 7 )

Threshold n = 100 n = 200 n = 500 n = 1000

u adm (cm) P f P f P f P f β β β β

12 1.44 ⋅ 10 − 1

[CoV = 0.08%]

1.06 1.45 ⋅ 10 − 1

[CoV = 0.08%]

1.06 1.44 ⋅ 10 − 01

[CoV = 0.08%]

1.06 1.44 ⋅ 10 − 1

[CoV = 0.08%]

1.06

15 1.33 ⋅ 10 − 2

[CoV = 0.27%]

2.22 1.44 ⋅ 10 − 2

[CoV = 0.26%]

2.19 1.45 ⋅ 10 − 02

[CoV = 0.26%]

2.18 1.45 ⋅ 10 − 2

[CoV = 0.26%]

2.18

20 3.85 3.69 3.66 3.66

5.84 ⋅ 10 − 5
[CoV = 4.14%]

1.13 ⋅ 10 − 4
[CoV = 2.97%]

1.26 ⋅ 10 − 04
[CoV = 2.82%]

1.24 ⋅ 10 − 4
[CoV = 2.84%]

21 4.15 3.97 3.90 3.91

1.66 ⋅ 10 − 5
[CoV = 7.76%]

3.64 ⋅ 10 − 5
[CoV = 5.24%]

4.72 ⋅ 10 − 05
[CoV = 4.60%]

4.56 ⋅ 10 − 5
[CoV = 4.68%]