Uncertain Soil Simulations
Motivation
Summary
Seismic Wave Propagation in Stochastic Soils
Boris Jeremić and Kallol Sett
Department of Civil and Environmental Engineering
University of California,
Davis, California, U.S.A.
4th ICEGE,
Thessaloniki, Greece,
June, 2007
Jeremić, Sett
Seismic Wave Propagation in Stochastic Soils
Computational Geomechanics Group
Motivation
Uncertain Soil Simulations
Summary
Outline
Motivation
Soils Behavior is Uncertain
Uncertain Soil Simulations
Probabilistic Soil Elasto–Plasticity
Stochastic Soil Dynamics
Summary
Jeremić, Sett
Seismic Wave Propagation in Stochastic Soils
Computational Geomechanics Group
Motivation
Uncertain Soil Simulations
Summary
Soils Behavior is Uncertain
Soils are Inherently Uncertain
Spatial Variation of Friction Angle
(Mayne et al. (2000))
Typical COVs of Different Soil Properties
(Lacasse and Nadim 1996)
Jeremić, Sett
Seismic Wave Propagation in Stochastic Soils
Computational Geomechanics Group
Uncertain Soil Simulations
Motivation
Summary
Soils Behavior is Uncertain
Characterization and Quantification
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Natural variability of soil deposit (Phoon and Kulhawy
1999, Fenton 1999) → function of soil formation process
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Testing error (Phoon and Kulhawy 1999, Marosi and
Hiltunen 2004, Stokoe et al. 2004)
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Ergodic assumption might not strictly apply (Der Kiureghian
2005)
Imperfection of instruments
Error in methods to register quantities
Transformation error (Phoon and Kulhawy 1999)
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Correlation by empirical data fitting (e.g. CPT data →
friction angle etc.)
Jeremić, Sett
Seismic Wave Propagation in Stochastic Soils
Computational Geomechanics Group
Uncertain Soil Simulations
Motivation
Summary
Probabilistic Soil Elasto–Plasticity
Our Original Developments
Constitutive : Second-order accurate (exact mean and
variance) PDF of stress-strain response (Fokker –
Planck – Kolmogorov Equation)
Spatial : Spectral Stochastic Elastic – Plastic Finite
Element Method, to simulate uncertain spatial
variability of elastic–plastic soils
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Obtain complete probabilistic description (PDF) for:
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Stresses–Strain response
Displacements (velocities, accelerations)
Use for:
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Sensitivity analysis
Probability of failure (tails of PDF)
Probabilistic site characterization design
Jeremić, Sett
Seismic Wave Propagation in Stochastic Soils
Computational Geomechanics Group
Uncertain Soil Simulations
Motivation
Summary
Probabilistic Soil Elasto–Plasticity
Input Soil Parameters Random Fields
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Truncated Karhunen–Loevé (KL) expansion
Representation of input random fields in eigen-modes of
covariance kernel
Su (x, θ) = S̄u (x) +
PM
n=1
√
λn ξn (θ)fn (x)
R
C(x1 , x2 )f (x2 )dx2 = λf (x1 )
Z
1
[Su (x, θ) − S̄u (x)]fi (x)dx
ξi (θ) = √
λi D
Error minimizing property
Optimal expansion → minimization of number of stochastic
dimensions
D
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Jeremić, Sett
Seismic Wave Propagation in Stochastic Soils
Computational Geomechanics Group
Uncertain Soil Simulations
Motivation
Summary
Probabilistic Soil Elasto–Plasticity
Stochastic Elastic–Plastic Finite Elements
N
X
e,ep
Kmn
dni +
N X
P
X
dnj
Z
Kmn =
Bn DBm dV
0
e,ep
Cijk Kmnk
= hFm ψi [{ζr }]i
k =1
n=1 j=0
n=1
M
X
;
0
Kmnk
D
Cijk = ζk (θ)ψi [{ζr }]ψj [{ζr }]
√
Bn Zλk hk Bm dV
D
Fm =
φNm dV
Z
=
;
D
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Based on SSFEM (Ghanem and Spanos (2003))
Random material and random forcing
Efficient representation of input random fields into finite
number of random variables using KL-expansion
Representation of (unknown) solution random variables
using polynomial chaos of (known) input random variables
Jeremić, Sett
Seismic Wave Propagation in Stochastic Soils
Computational Geomechanics Group
Uncertain Soil Simulations
Motivation
Summary
Probabilistic Soil Elasto–Plasticity
Probabilistic Elasto–Plasticity
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Probabilistic elastic–plastic constitutive incremental
e,ep
∆kl
equation ∆σij = Dijkl
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e,ep
Random stiffness Dijkl
Random strain increment ∆kl
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Use of Euler Lagrange form of Fokker–Planck–Kolmogorov
(FPK) equation (Kavvas 2003) to obtain
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Second-order accurate (exact mean and variance)
stress–strain solution (PDF)
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Complete probabilistic description of response → PDF
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FPK equation is applicable to any elastic–plastic material
model
Jeremić, Sett
Seismic Wave Propagation in Stochastic Soils
Computational Geomechanics Group
Uncertain Soil Simulations
Motivation
Summary
Probabilistic Soil Elasto–Plasticity
1–D Low and High OCR Cam Clay
0
Strain (%)
0
1.62
Strain (%)
2.0
0.03
0.04
Mode Line
0.025
Std. Deviation Lines
Mean Line
0.02
Stress (MPa)
Stress (MPa)
0.03
50
0.02
30
0.015
0.01
10
0.01
Std. Deviation Lines
Deterministic Line
100
Mean Line
0.005
50
Deterministic Line
30
Mode Line
0
0
0.05
0.1
0.15
0.2
Time (s)
0.25
0.3
0
0
0.05
0.1
0.15
0.2
Time (s)
0.25
0.3
random G, M and p0
Jeremić, Sett
Seismic Wave Propagation in Stochastic Soils
Computational Geomechanics Group
0.37
Uncertain Soil Simulations
Motivation
Summary
Stochastic Soil Dynamics
1–D Shear Column Example
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Static pushover of a
Stochastic
shear column
10m high (deep)
Small correlation
length results in
mean that tends to
deterministic solution
High correlation
length increases
mean and
standard deviation
Jeremić, Sett
Seismic Wave Propagation in Stochastic Soils
Computational Geomechanics Group
Uncertain Soil Simulations
Motivation
Summary
Stochastic Soil Dynamics
Stochastic Seismic Ground Motions
1
Displacement (m)
8
6
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1
2
3
4
5
Time (s)
6
7
8
2
5
10
Seismic Wave Propagation in Stochastic Soils
20
Time (s)
−1
0.3
0.25
0.2
0.15
Sinusoidal motions example
0.1
0.05
Complete PDF of motions
for each time step.
In general, increase in system uncertainty
Jeremić, Sett
15
−0.5
Std. Dev. of Displacement (m)
1
0
ent
0
0
lace
m
−2
−1
2
0.5
(m)
4
Disp
Probability Densities of Displacement
10
5
10
15
20
Time (s)
Computational Geomechanics Group
Uncertain Soil Simulations
Motivation
Summary
Stochastic Soil Dynamics
Stochastic Seismic Ground Motions
Displacement [m]
0.002
0.001
1
2
3
4
5
Time [s]
−0.001
−0.002
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Example motions: Imperial Valley
Mean and SD of ground motions
Large uncertainty at ground motion peaks
Jeremić, Sett
Seismic Wave Propagation in Stochastic Soils
Computational Geomechanics Group
Uncertain Soil Simulations
Motivation
Summary
Summary
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A new, second-order accurate (exact mean and variance)
formulation for probabilistic elastic–plastic soil simulation
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Analytic modeling and simulations of
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spatial variability and
point-wise uncertainty
of soil elastic of elastic–plastic properties for static and
dynamic problems
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Application to:
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Sensitivity analysis
Probability of failure (tails of PDF)
Site characterization design (probabilistic)
General stochastic modeling in elastic–plastic solid and
structural mechanics
Jeremić, Sett
Seismic Wave Propagation in Stochastic Soils
Computational Geomechanics Group