Polynomial chaos expansions and stochastic finite-element methods - Risk and Reliability in Geotechnical Engineering - page 283

Consequently, due to the orthogonality of the PC basis, the partial variance D u reduces to

= ∑

PC

2

D = Var M () X y .

(6.75)

u u u

α∈ A

u

In other words, from a given PC expansion, the Sobol' indices at any order may be
obtained by a mere combination of the squares of the coefficients. As an illustration, the
first-order PC-based Sobol' indices read:

∑

PC 2

S = y D where A = { α ∈ A : α > 0 , α = 0 },

(6.76)

i α i i j ≠ i

α∈A

i

whereas the total PC-based Sobol' indices are

∑ α

T , PC 2 T

S = y D where A = { α ∈ A : α > 0 }.

i i i

(6.77)

i T

α∈A

PCEs and the various types of post-processing presented above are now applied to differ-
ent classical geotechnical problems.

6.5 aPPlICatIon eXaMPleS

6.5.1 load-carrying capacity of a strip footing
6.5.1.1 Independent input variables
Let us consider the strip footing of width B = 10 m sketched in Figure 6.6 that is embedded
at depth D . We assume that the ground water table is far below the surface. The soil layer is
assumed homogeneous with cohesion c , friction angle ϕ, and unit weight γ.
The ultimate bearing capacity reads (Lang et al., 2007):

1
2

(6.78)

qcNDN = + γ + γ N ,

u c q γ

B

σ f

γ D

c , γ , ϕ

Figure 6.6 Example #1: Strip footing.