Application of the subset simulation approach to spatially varying soils - Risk and Reliability in Geotechnical Engineering - page 556

Table 15.2 Eigenvalues and eigenvectors of a 2D random field, where μ = 60 MPa, COV = 15%,
l lln x = l lln y = 1 m, a x = 7.5 m, a y = 3 m, and M = 2

y y

λ j x

λ y λλλ

2 = D

j x

φ j x ( x = 05m . )

φ k y

2 =

i D j x j k ( y = 3m )

φφφ

i

k k

1 1 1.93378 1.72722 3.34008 0.34220 0.18802 0.06434

1 2 1.93378 1.19752 2.31575 0.34220 0.33382 0.11423

2 1 1.75749 1.72722 3.03557 0.06380 0.18802 0.01199

2 2 1.75749 1.19752 2.10465 0.06380 0.33382 0.02130

Table 15.3 Retained eigenvalues and eigenvectors of a 2D
random field where μ = 60 MPa, COV = 15% , l lln
x = l lln y = 1 m, a x = 7.5 m, a y = 3 m, and M = 2

i

2 D

λ i D

2

φ i

1 3.34008 0.06434

2 3.03557 0.01199

The rows of Table 15.2 should now be arranged according to a decreasing order of the
eigenvalues λ i 2D . Finally, the highest M eigenvalues and the corresponding eigenfunctions
are retained ( Table 15.3 ). These values are then substituted in the K-L expansion to calcu-
late the value of the 2D random field at a given point. For a given realization (say ξ 1 = 0.815,
ξ 2 = 0.906), the value of the Young's modulus at the arbitrary point ( x = 0.5 m and y = 3 m)
as given by Equation 15.11 is equal to E = 60,360,433 Pa.
The accuracy of the discretized random field depends on the size of the K-L expansion
(i.e., the number of terms M ). Figure 15.2 presents the error estimate of the discretized 2D
random field versus the number of eigenmodes (i.e., the number of terms of K-L expan-
sion) for two values of the autocorrelation distance of an isotropic soil ( l lln x = l lln y = 1 m
and l lln x = l lln y = 3 m). For a prescribed accuracy, a greater number of terms is required for a
smaller value of the autocorrelation distance. Figure 15.2 also indicates (as expected) that
a more accurate representation of the target random field is obtained for a greater number

50

l lln x = l lln y = 1 m
l lln x = l lln y = 3 m

45

40

35

30

25

20

15

10

5

0

0 200 400 600 800 1000 1200 1400 1600 1800 2000

M

Figure 15.2 Error estimate versus the number of eigenmodes for two values of the autocorrelation distance
of an isotropic soil.