r - The Hong Kong Polytechnic University

RANDOM FIELD AND SPATIAL AVERAGING
Stationary random fields are widely used primarily because the amount of measured data available in most site
investigations would only permit characterization under this second-moment (weak) stationarity assumption. A
two dimensional stationary random field for shear strength τ f (x,z) can be characterized by its point mean value
E(τ f ), point variance Var(τ f ), and auto-correlation function. The auto-correlation function of a stationary random
field τ f (x,z) is defined as the correlation between two locations with lag distance of Δx and Δz:
COV ( τ
f
( x, z), τ
f
( x +Δ x, z +Δz))
ρ( Δx, Δz) ≡ ρ( τ
f
( x, z), τ
f
( x+Δ x, z+Δ z))
=
(1)
Var( τ ( x, z)) ⋅ Var( τ ( x +Δ x, z +Δz))
where Var denotes variation; COV denotes covariance. An auto-correlation model widely used in geotechnical
engineering literature is the single exponential model:
ρ( Δx, Δ z) = exp( −k Δx −k Δ z)
(1)
x
z
where the parameter k x and k z are respectively equal to 2 divided by the scales of fluctuation (SOF) in the x and
z directions, denoted by SOF x and SOF z . It is clear that the correlation decreases as Δx and Δz increase. This is
consistent with measurements taken from natural soils: soil properties are strongly correlated within a small
interval but are weakly correlated over a large interval. The SOF is the correlation length, i.e. the length scale
within which two locations are significantly correlated.
Vanmarcke (1977) pointed out that the spatial average of soil properties over a region D has a mean value
identical to the point mean but has a variance smaller than the point variance. Let the region D be a rectangular
domain defined by [x 0 x 0 +Δx] and [z 0 z 0 +Δz]. Mathematically, the spatial average over D can be defined as
τ
D
f
1
x z
x0+Δ x z0+Δz
= ΔΔ
∫ ∫
x0 z0
τ
f
( , )
x z dzdx
The variance of τ D f is smaller than the point variance Var(τ f ) due to the cancellation of variability through spatial
D
averaging. Vanmarcke (1977) further defined a variance reduction factor which is equal to the variance of τ f
divided by the point variance:
2 D
(4)
Γ ( D)
= Var( τ
f ) Var( τ
f )
Using the single exponential model in Eq. (2), Vanmarcke (1977) showed that
2 2
⎡
2 2Δx ⎛ 2Δx ⎞⎤ 2Δx ⎡ 2Δz ⎛ 2Δz ⎞⎤
2Δz
Γ ( D) = ⎢ − 1+ exp⎜− ⎟⎥ × 1 exp
2 ⎢ − + ⎜−
⎟⎥
2
(5)
⎣SOFx ⎝ SOFx ⎠⎦
SOFx
⎣SOFz ⎝ SOFz
⎠⎦
SOFz
Γ 2 (D) is a decreasing function of Δx/SOFx and Δz/SOFz. These latter two terms may be interpreted as the
numbers of independent segments in the x and z directions.
COMPARISON BETWEEN EFFECTIVE τ f AND ITS SPATIAL AVERAGING
Although Vanmarcke’s theory provides useful closed-form solutions for spatial average, it is not clear if the
effective shear strength is the same as τ f D in the first instance. To verify this, random field finite element analyses
(FEA) are conducted. The region D is taken to be a plane-strain 36mx10m rectangular area (Δx = 10m, Δz = 36m)
with 0.1m×0.1m FEA mesh grids. The total number of plane-strain elements is 36,000. The two lateral boundaries
are free, the bottom boundary is roller, and the lower-left-most node is a hinge. The upper boundary is subjected to
a compression stress. The unit weights of all elements are zeros, the Young’s modulus is 40 MN/m 2 , and the
Poisson ratio is 0.3. The elastic properties hardly affect the failure load and a high Young’s modulus is selected for
computational efficiency.
The spatially varying shear strength τ f is simulated by stationary Gaussian random fields with a point mean E(τ f ) =
50 kN/m 2 , with a point standard deviation Var(τ f ) 0.5 = 10 kN/m 2 , and with SOFx = SOFz = SOF. When assigning
the simulated τ f to each element, the local averaging subdivision algorithm developed by Fenton and Vanmarcke
(1990) is taken to conduct local averaging within each 0.1m×0.1m element. In this initial study, the shear strength
τ f is assumed to be independent of the confining pressure for simplicity, i.e., φ = 0 o .
f
f
(3)
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