Polynomial chaos expansions and stochastic finite-element methods - Risk and Reliability in Geotechnical Engineering - page 284

Table 6.3 Ultimate bearing capacity of a strip foundation—probabilistic model
Parameter

Notation Type of PDF Mean value Coefficient of variation

Foundation width B Deterministic 10 m -

Foundation depth D Gaussian 1 m 15%

Unit soil weight Lognormal 20 kN/m 3 10% γ

Cohesion c Lognormal 20 kPa 25%

Friction angle Beta 10%

Range: [0, 45°], μ = 30° ϕ

where the bearing capacity factors read:

N q = e π tanϕ tan 2 (π/4 + ϕ/2).

N c = ( N q − 1) cot ϕ. (6.79)

N γ = 2( N q − 1) tan ϕ.

The soil parameters and the foundation depth are considered as independent random
variables, whose properties are listed in Table 6.3 . Let us denote the model input vector by
X = { D , γ, c , ϕ} T . The associated random-bearing capacity is q u ( X ).
Using the m ean values of the parameters in Table 6.3 , the ultimate bearing capacity
is equal l to q u

= 2.78MPa. We now consider several design situations with applied loads

des u = where SF = 1.5, 2, 2.5, and 3 would be the global safety factor obtained from
a deterministic design. Then we consider the reliability of the foundation with respect to the
ultimate bearing capacity. The limit state function reads:

q q SF

g () X = q () X − q = q () X − qSF . (6.80)

u des u u

Classical reliability methods are used, namely FORM, SORM, and crude MCS with 10 7
samples to get a reference solution. The uncertainty quantification software UQLab is used
(Marelli and Sudret, 2014). Alternatively, a PC expansion q PC ( X of the ultimate bearing
capacity is first computed using an LHS ED of size n = 500. Then the PC expansion is sub-
stituted for in Equation 6.80 and the associated probability of failure is computed by MCS
(10 7 samples), now using only the PC expansion (and for the different values of SF). The
results are reported in Table 6.4 .
From Table 6.4 , it is clear that the results obtained by PC expansion are almost equal to
those obtained by the reference MCS. The relative error in terms of the probability of failure
is all in all <1% (the corresponding error on the generalized reliability index β gen = −Ф −1 ( P f )
is negligible).

Table 6.4 Ultimate bearing capacity of a strip foundation—probability of failure (resp. generalized
reliability index β gen = −Ф − 1 ( P f ) between parentheses)—case of independent variables

SF FORM SORM MCS (10 7 Runs) PCE + MCS a

1.5

1.73 ⋅ 10 − 1 (0.94) 1.70 ⋅ 10 − 1 (0.96) 1.69 ⋅ 10 − 1 (0.96) 1.70 ⋅ 10 − 1 (0.96)

2.0

5.49 ⋅ 10 − 2 (1.60) 5.30 ⋅ 10 − 2 (1.62) 5.30 ⋅ 10 − 2 (1.62) 5.29 ⋅ 10 − 2 (1.62)

2.5 1.72 ⋅ 10 − 2 (2.11) 1.65 ⋅ 10 − 2 (2.13) 1.63 ⋅ 10 − 2 (2.14) 1.65 ⋅ 10 − 2 (2.13)

3.0

5.54 ⋅ 10 − 3 (2.54) 5.23 ⋅ 10 − 3 (2.56) 5.24 ⋅ 10 − 3 (2.56) 5.20 ⋅ 10 − 3 (2.56)

n = 500, n MCS = 10 7 .

a