The Case for Probabilistic Elasto--Plasticitysokocalo.engr.ucdavis.edu/~jeremic/ · (After Lacasse...
Transcript of The Case for Probabilistic Elasto--Plasticitysokocalo.engr.ucdavis.edu/~jeremic/ · (After Lacasse...
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
The Case for Probabilistic Elasto–Plasticity
Boris JeremicKallol Sett (UA) and Lev Kavvas
Department of Civil and Environmental EngineeringUniversity of California, Davis
GheoMatMasseria Salamina
Italy, June 2009
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
OutlineMotivation
Stochastic Systems: Historical PerspectivesUncertainties in Material
Probabilistic Elasto–PlasticityPEP FormulationsProbabilistic Elastic–Plastic Response
Stochastic Elastic–Plastic Finite Element MethodSEPFEM FormulationsSEPFEM Verification Example
ApplicationsSeismic Wave Propagation Through Uncertain SoilsProbabilistic Analysis for Decision Making
Summary
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Stochastic Systems: Historical Perspectives
OutlineMotivation
Stochastic Systems: Historical PerspectivesUncertainties in Material
Probabilistic Elasto–PlasticityPEP FormulationsProbabilistic Elastic–Plastic Response
Stochastic Elastic–Plastic Finite Element MethodSEPFEM FormulationsSEPFEM Verification Example
ApplicationsSeismic Wave Propagation Through Uncertain SoilsProbabilistic Analysis for Decision Making
Summary
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Stochastic Systems: Historical Perspectives
Failure Mechanisms for Geomaterials
Soil: Inside Failure of "Uniform" MGM Specimen(After Swanson et al. 1998)Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Stochastic Systems: Historical Perspectives
Personal Motivation
I Probabilistic fish counting
I Williams’ DEM simulations, differential displacementvortices
I SFEM round table
I Kavvas’ probabilistic hydrology
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Stochastic Systems: Historical Perspectives
Types of UncertaintiesI Epistemic uncertainty - due to lack of knowledge
I Can be reduced by collecting more dataI Mathematical tools are not well developedI trade-off with aleatory uncertainty
I Aleatory uncertainty - inherent variation of physical systemI Can not be reducedI Has highly developed mathematical tools
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Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Stochastic Systems: Historical Perspectives
ErgodicityI Exchange ensemble averages for time averages
I Is soil elasto-plasticity ergodic?I Can soil elastic–plastic statistical properties be obtained by
temporal averaging?I Will soil elastic–plastic statistical properties "renew" at each
occurrence?I Are soil elastic–plastic statistical properties statistically
independent?
I Claim in literature that structural nonlinear behavior isnon–ergodic while earthquake characteristics are (?!)
I However, earthquake characteristics is representingmechanics (fault slip) on a different scale...
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Stochastic Systems: Historical Perspectives
Historical OverviewI Brownian motion, Langevin equation → PDF governed by
simple diffusion Eq. (Einstein 1905)
I With external forces → Fokker-Planck-Kolmogorov (FPK)for the PDF (Kolmogorov 1941)
I Approach for random forcing → relationship between theautocorrelation function and spectral density function(Wiener 1930)
I Approach for random coefficient → Functional integrationapproach (Hopf 1952), Averaged equation approach(Bharrucha-Reid 1968), Numerical approaches, MonteCarlo method
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Uncertainties in Material
OutlineMotivation
Stochastic Systems: Historical PerspectivesUncertainties in Material
Probabilistic Elasto–PlasticityPEP FormulationsProbabilistic Elastic–Plastic Response
Stochastic Elastic–Plastic Finite Element MethodSEPFEM FormulationsSEPFEM Verification Example
ApplicationsSeismic Wave Propagation Through Uncertain SoilsProbabilistic Analysis for Decision Making
Summary
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Uncertainties in Material
Material Behavior Inherently Uncertain
I Spatialvariability
I Point-wiseuncertainty,testingerror,transformationerror
(Mayne et al. (2000)
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Uncertainties in Material
Motivation
Typical Coefficients of Variation of Different Soil Properties
(After Lacasse and Nadim 1996)Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Uncertainties in Material
Soil Uncertainties and Quantification
I Natural variability of soil deposit (Fenton 1999)I Function of soil formation process
I Testing error (Stokoe et al. 2004)I Imperfection of instrumentsI Error in methods to register quantities
I Transformation error (Phoon and Kulhawy 1999)I Correlation by empirical data fitting (e.g. CPT data →
friction angle etc.)
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Uncertainties in Material
Probabilistic Material (Soil Site) Characterization
I Ideal: complete probabilistic site characterizationI Large (physically large but not statistically) amount of data
I Site specific mean and coefficient of variation (COV)I Covariance structure from similar sites (e.g. Fenton 1999)
I Moderate amount of data → Bayesian updating (e.g.Phoon and Kulhawy 1999, Baecher and Christian 2003)
I Minimal data: general guidelines for typical sites and testmethods (Phoon and Kulhawy (1999))
I COVs and covariance structures of inherent variabilityI COVs of testing errors and transformation uncertainties.
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Uncertainties in Material
Recent State-of-the-ArtI Governing equation
I Dynamic problems → Mu + Cu + Ku = φI Static problems → Ku = φ
I Existing solution methodsI Random r.h.s (external force random)
I FPK equation approachI Use of fragility curves with deterministic FEM (DFEM)
I Random l.h.s (material properties random)I Monte Carlo approach with DFEM → CPU expensiveI Perturbation method → a linearized expansion! Error
increases as a function of COVI Spectral method → developed for elastic materials so far
I New developments for elasto–plastic applications
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
PEP Formulations
OutlineMotivation
Stochastic Systems: Historical PerspectivesUncertainties in Material
Probabilistic Elasto–PlasticityPEP FormulationsProbabilistic Elastic–Plastic Response
Stochastic Elastic–Plastic Finite Element MethodSEPFEM FormulationsSEPFEM Verification Example
ApplicationsSeismic Wave Propagation Through Uncertain SoilsProbabilistic Analysis for Decision Making
Summary
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
PEP Formulations
Uncertainty Propagation through Constitutive Eq.
I Incremental el–pl constitutive equationdσij
dt= Dijkl
dεkl
dt
Dijkl =
Del
ijkl for elastic
Delijkl −
DelijmnmmnnpqDel
pqkl
nrsDelrstumtu − ξ∗r∗
for elastic–plastic
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
PEP Formulations
Previous WorkI Linear algebraic or differential equations → Analytical
solution:I Variable Transf. Method (Montgomery and Runger 2003)I Cumulant Expansion Method (Gardiner 2004)
I Nonlinear differential equations(elasto-plastic/viscoelastic-viscoplastic):
I Monte Carlo Simulation (Schueller 1997, De Lima et al2001, Mellah et al. 2000, Griffiths et al. 2005...)→ accurate, very costly
I Perturbation Method (Anders and Hori 2000, Kleiber andHien 1992, Matthies et al. 1997)→ first and second order Taylor series expansion aboutmean - limited to problems with small C.O.V. and inherits"closure problem"
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
PEP Formulations
Problem Statement
I Incremental 3D elastic-plastic stress–strain:
dσij
dt=
{Del
ijkl −Del
ijmnmmnnpqDelpqkl
nrsDelrstumtu − ξ∗r∗
}dεkl
dt
I Focus on 1D → a nonlinear ODE with random coefficient(material) and random forcing (ε)
dσ(x , t)dt
= β(σ(x , t),Del(x),q(x), r(x); x , t)dε(x , t)
dt= η(σ,Del ,q, r , ε; x , t)
with initial condition σ(0) = σ0
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
PEP Formulations
Evolution of the Density ρ(σ, t)
I From each initial point inσ-space a trajectorystarts out describingthe corresponding solutionof the stochastic process
I Movement of a cloud of initialpoints described by densityρ(σ,0) in σ-space,is governed by theconstitutive equation,
t
P( )
σσ
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
PEP Formulations
Stochastic Continuity (Liouville) Equation
I phase density ρ of σ(x , t) varies in time according to acontinuity Liouville equation (Kubo 1963):
∂ρ(σ(x , t), t)∂t
=
−∂η(σ(x , t),Del(x),q(x), r(x), ε(x , t))∂σ
ρ[σ(x , t), t ]
I with initial conditions ρ(σ,0) = δ(σ − σ0)
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
PEP Formulations
Ensemble Average form of Liouville EquationContinuity equation written in ensemble average form (eg.cumulant expansion method (Kavvas and Karakas 1996)):
∂ 〈ρ(σ(xt , t), t)〉∂t
= − ∂
∂σ
»fiη(σ(xt , t), Del(xt), q(xt), r(xt), ε(xt , t))
fl+
Z t
0dτCov0
»∂η(σ(xt , t), Del(xt), q(xt), r(xt), ε(xt , t))
∂σ;
η(σ(xt−τ , t − τ), Del(xt−τ ), q(xt−τ ), r(xt−τ ), ε(xt−τ , t − τ)
–ff〈ρ(σ(xt , t), t)〉
–+
∂2
∂σ2
»Z t
0dτCov0
»η(σ(xt , t), Del(xt), q(xt), r(xt), ε(xt , t));
η(σ(xt−τ , t − τ), Del(xt−τ ), q(xt−τ ), r(xt−τ ), ε(xt−τ , t − τ))
– ff〈ρ(σ(xt , t), t)〉
–
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
PEP Formulations
Eulerian–Lagrangian FPK Equationvan Kampen’s Lemma (van Kampen 1976) → < ρ(σ, t) >= P(σ, t),ensemble average of phase density is the probability density;
∂P(σ(xt , t), t)∂t
= − ∂
∂σ
»fiη(σ(xt , t), Del(xt), q(xt), r(xt), ε(xt , t))
fl+
Z t
0dτCov0
»∂η(σ(xt , t), Del(xt), q(xt), r(xt), ε(xt , t))
∂σ;
η(σ(xt−τ , t − τ), Del(xt−τ ), q(xt−τ ), r(xt−τ ), ε(xt−τ , t − τ)
–ffP(σ(xt , t), t)
–+
∂2
∂σ2
»Z t
0dτCov0
»η(σ(xt , t), Del(xt), q(xt), r(xt), ε(xt , t));
η(σ(xt−τ , t − τ), Del(xt−τ ), q(xt−τ ), r(xt−τ ), ε(xt−τ , t − τ))
– ffP(σ(xt , t), t)
–
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
PEP Formulations
E–L FPK Equation
I Advection-diffusion equation
∂P(σ, t)∂t
= − ∂
∂σ
[N(1)P(σ, t)− ∂
∂σ
{N(2)P(σ, t)
}]I Complete probabilistic description of responseI Solution PDF is second-order exact to covariance of time
(exact mean and variance)I It is deterministic equation in probability density spaceI It is linear PDE in probability density space → simplifies
the numerical solution process
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
PEP Formulations
Template Solution of FPK EquationI FPK diffusion–advection equation is applicable to any
material model → only the coefficients N(1) and N(2) aredifferent for different material models
∂P(σ, t)∂t
= − ∂
∂σ
[N(1)P(σ, t)− ∂
∂σ
{N(2)P(σ, t)
}]= −∂ζ
∂σ
I Initial conditionI Deterministic → Dirac delta function → P(σ,0) = δ(σ)I Random → Any given distribution
I Boundary condition: Reflecting BC → conservesprobability mass ζ(σ, t)|At Boundaries = 0
I Finite Differences used for solution (among many others)
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
PEP Formulations
Application of FPK equation to Material Models
I FPK equation is applicable to any incrementalelastic–plastic material model
I Solution in terms of PDF, not a single value of stress
I Influence of initial condition on the PDF of stress
I Mean stress yielding or
I Probabilistic yielding
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Probabilistic Elastic–Plastic Response
OutlineMotivation
Stochastic Systems: Historical PerspectivesUncertainties in Material
Probabilistic Elasto–PlasticityPEP FormulationsProbabilistic Elastic–Plastic Response
Stochastic Elastic–Plastic Finite Element MethodSEPFEM FormulationsSEPFEM Verification Example
ApplicationsSeismic Wave Propagation Through Uncertain SoilsProbabilistic Analysis for Decision Making
Summary
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Probabilistic Elastic–Plastic Response
Elastic Response with Random GI General form of elastic constitutive rate equation
dσ12
dt= 2G
dε12
dt= η(G, ε12; t)
I Advection and diffusion coefficients of FPK equation
N(1) = 2dε12
dt< G >
N(2) = 4t(
dε12
dt
)2
Var [G]
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Probabilistic Elastic–Plastic Response
Elastic Response with Random G
0.002
0.004
0.006
0
0.001
0.002
0.003
0.0040
2500
5000
7500
10000
0.00789
Strain (%)
0.0027
0.0426
Stress (MPa)Time (Sec)
Prob Density
0.002
0.004
0.006
< G > = 2.5 MPa; Std. Deviation[G] = 0.5 MPaJeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Probabilistic Elastic–Plastic Response
Verification – Variable Transformation Method
0.002 0.004 0.006 0.008
0.0005
0.001
0.0015
0.002
0.0025
Strain (%)0 0.0426
Std. Deviation Lines
Std. Deviation Lines
(Variable Transformation)
(Fokker−Planck)
(Fokker−Planck)
(Variable Transformation)Mean Line
Mean Line
Time (Sec)
Stre
ss (M
Pa)
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Probabilistic Elastic–Plastic Response
Modified Cam Clay Constitutive Modeldσ12
dt= Gep dε12
dt= η(σ12,G,M,e0,p0, λ, κ, ε12; t)
η =
2G −
(36
G2
M4
)σ2
12
(1 + e0)p(2p − p0)2
κ+
(18
GM4
)σ2
12 +1 + e0
λ− κpp0(2p − p0)
Advection and diffusion coefficients of FPK equation
N(i)(1) =
⟨η(i)(t)
⟩+
∫ t
0dτcov
[∂η(i)(t)∂t
; η(i)(t − τ)
]
N(i)(2) =
∫ t
0dτcov
[η(i)(t); η(i)(t − τ)
]Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Probabilistic Elastic–Plastic Response
Low OCR Cam Clay with Random G, M and p0
I Non-symmetry inprobabilitydistribution
I Differencebetweenmean, mode anddeterministic
I Divergence atcritical statebecause M isuncertain
0 0.05 0.1 0.15 0.2 0.25 0.3
0
0.01
0.02
0.03
0.04
Time (s)
Strain (%)0 1.62
Std. Deviation Lines
Mean LineDeterministic Line
Mode Line
Stre
ss (M
Pa)
50
30
10
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Probabilistic Elastic–Plastic Response
Comparison of Low OCR Cam Clay at ε = 1.62 %
0.01 0.02 0.03 0.04
100
200
300
400
500
Det
. Val
ue =
0.0
2906
MPa
Shear Stress (MPa)
of S
hear
Str
ess
Random G, M, p0Random G, M
Random G, p0
Random G
I None coincides with deterministicI Some very uncertain, some very certainI Either on safe or unsafe side
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Probabilistic Elastic–Plastic Response
High OCR Cam Clay with Random G and M
I Large non-symmetryin probabilitydistribution
I Significantdifferences inmean, mode,and deterministic
I Divergence atcritical state,M is uncertain 0 0.05 0.1 0.15 0.2 0.25 0.3 0.37
0
0.005
0.01
0.015
0.02
0.025
0.03
Strain (%)
Mean Line
Std. Deviation Lines
Time (s)
0 2.0
Stre
ss (M
Pa)
3050
100
Deterministic LineMode Line
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
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Probabilistic Elastic–Plastic Response
Probabilistic Yielding
I Weighted elastic and elastic–plastic Solution∂P(σ, t)/∂t = −∂
(Nw
(1)P(σ, t)− ∂(
Nw(2)P(σ, t)
)/∂σ
)/∂σ
I Weighted advection and diffusion coefficients are thenNw
(1,2)(σ) = (1− P[Σy ≤ σ])Nel(1) + P[Σy ≤ σ]Nel−pl
(1)
I Cumulative Probability Density function (CDF) of the yieldfunction
0.00005 0.0001 0.00015Σy HMPaL
0.2
0.4
0.6
0.8
1F@SyD = P@Sy£ΣyD
(a)(b)
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Probabilistic Elastic–Plastic Response
Transformation of a Bi–Linear (von Mises) Response
0 0.0108 0.0216 0.0324 0.0432 0.0540
0.00005
0.0001
0.00015
0.0002
0.00025
0.0003
Stre
ss (
MPa
)
Strain (%)
Mode
DeterministicSolutionStd. Deviations
Mean
linear elastic – linear hardening plastic von MisesJeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Probabilistic Elastic–Plastic Response
SPT Based Determination of Shear Strength
10 20 30 40
100
200
300
400
500
SPT N value
Su = (101.125*0.29) N 0.72
Und
rain
ed S
hear
Str
engt
h (k
Pa)
−300 −200 −100 0 100 200
0.002
0.004
0.006
0.008
0.01
Nor
mal
ized
Fre
quen
cy
Residual (w.r.t Mean) Undrained Shear Strength (kPa)
Transformation relationship between SPT N-value andundrained shear strength, su (cf. Phoon and Kulhawy (1999B)Histogram of the residual (w.r.t the deterministic transformationequation) undrained strength, along with fitted probabilitydensity function
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
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Probabilistic Elastic–Plastic Response
SPT Based Determination of Young’s Modulus
5 10 15 20 25 30 35
5000
10000
15000
20000
25000
30000
SPT N Value
You
ng’s
Mod
ulus
, E (
kPa)
E = (101.125*19.3) N 0.63
−10000 0 10000
0.00002
0.00004
0.00006
0.00008
Residual (w.r.t Mean) Young’s Modulus (kPa)
Nor
mal
ized
Fre
quen
cy
Transformation relationship between SPT N-value andpressure-meter Young’s modulus, E (cf. Phoon and Kulhawy(1999B))Histogram of the residual (w.r.t the deterministic transformationequation) Young’s modulus, along with fitted probability densityfunction
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
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Probabilistic Elastic–Plastic Response
Cyclic Response of Such Uncertain Material
−1 −0.5 0.5 1
−0.2
−0.1
0.1
0.2
Deterministic
Mean
She
ar S
tres
s (M
Pa)
Shear Strain (%)
−1 −0.5 0.5 1
0.05
0.1
0.15
0.2
Shear Strain (%)
She
ar S
tres
s (M
Pa)
Standard Deviation
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
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Probabilistic Elastic–Plastic Response
G/Gmax Response
0.01 0.02 0.05 0.1 0.2 0.5 1
0.2
0.4
0.6
0.8
1
1.2
Shear Strain (%)
PI=200% (Vucetic and Dobry 1991)
PI=100% (Vucetic and Dobry 1991)
PI=100% (Stokoe et al. 2004)Deterministic
Mean
G/G
max
Mean±Standard Deviation
Jeremic Computational Geomechanics Group
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Probabilistic Elastic–Plastic Response
Damping Response
0.01 0.02 0.05 0.1 0.2 0.5 1
5
10
15
20
Dam
ping
Rat
io (
%)
Shear Strain (%)
Corresponding to hysteresisloop of Mean of shear stress
Corresponding to hysteresis loop ofMean±Standard Deviation of shear stress
Deterministic
PI=100% (Stokoe et al. 2004)
PI=100% (Vucetic and Dobry 1991)
PI=200% (Vucetic and Dobry 1991)
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
SEPFEM Formulations
OutlineMotivation
Stochastic Systems: Historical PerspectivesUncertainties in Material
Probabilistic Elasto–PlasticityPEP FormulationsProbabilistic Elastic–Plastic Response
Stochastic Elastic–Plastic Finite Element MethodSEPFEM FormulationsSEPFEM Verification Example
ApplicationsSeismic Wave Propagation Through Uncertain SoilsProbabilistic Analysis for Decision Making
Summary
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
SEPFEM Formulations
Governing Equations & Discretization Scheme
I Governing equations in geomechanics:
Aσ = φ(t); Bu = ε; σ = Dε
I Discretization (spatial and stochastic) schemesI Input random field material properties (D) →
Karhunen–Loève (KL) expansion, optimal expansion, errorminimizing property
I Unknown solution random field (u) → Polynomial Chaos(PC) expansion
I Deterministic spatial differential operators (A & B) →Regular shape function method with Galerkin scheme
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Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
SEPFEM Formulations
Spectral Stochastic Elastic–Plastic FEM
I Minimizing norm of error of finite representation usingGalerkin technique (Ghanem and Spanos 2003):
N∑n=1
Kmndni +N∑
n=1
P∑j=0
dnj
M∑k=1
CijkK ′mnk = 〈Fmψi [{ξr}]〉
Kmn =
∫D
BnDBmdV K ′mnk =
∫D
Bn√λkhkBmdV
Cijk =⟨ξk (θ)ψi [{ξr}]ψj [{ξr}]
⟩Fm =
∫DφNmdV
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
SEPFEM Formulations
Inside SEPFEM
I Explicit stochastic elastic–plastic finite elementcomputations
I FPK probabilistic constitutive integration at Gaussintegration points
I Increase in (stochastic) dimensions (KL and PC) of theproblem
I Development of the probabilistic elastic–plastic stiffnesstensor
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
SEPFEM Verification Example
OutlineMotivation
Stochastic Systems: Historical PerspectivesUncertainties in Material
Probabilistic Elasto–PlasticityPEP FormulationsProbabilistic Elastic–Plastic Response
Stochastic Elastic–Plastic Finite Element MethodSEPFEM FormulationsSEPFEM Verification Example
ApplicationsSeismic Wave Propagation Through Uncertain SoilsProbabilistic Analysis for Decision Making
Summary
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
SEPFEM Verification Example
1–D Static Pushover Test Example
I Linear elastic model:< G >= 2.5 kPa,Var [G] = 0.15 kPa2,correlation length for G = 0.3 m.
I Elastic–plastic material model,von Mises, linear hardening,< G >= 2.5 kPa,Var [G] = 0.15 kPa2,correlation length for G = 0.3 m,Cu = 5 kPa,C
′u = 2 kPa.
����������������������
L
F
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
SEPFEM Verification Example
Linear Elastic FEM Verification
1 2 3 4 5
0.02
0.04
0.06
0.08
0.1
Loa
d (N
)
Displacement at Top Node (mm)
Mean (SFEM)
Standard Deviations (MC)
Mean (MC)
Standard Deviations (SFEM)
Mean and standard deviations of displacement at the top node,linear elastic material model,KL-dimension=2, order of PC=2.
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
SEPFEM Verification Example
SEPFEM verification
1 2 3 4 5 6 7
0.02
0.04
0.06
0.08
0.1L
oad
(N)
Displacement at Top Node (mm)
Standard Deviations (SFEM)
Standard Deviations (MC)
Mean (SFEM)
Mean (MC)
Mean and standard deviations of displacement at the top node,von Mises elastic-plastic linear hardening material model,KL-dimension=2, order of PC=2.
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Seismic Wave Propagation Through Uncertain Soils
OutlineMotivation
Stochastic Systems: Historical PerspectivesUncertainties in Material
Probabilistic Elasto–PlasticityPEP FormulationsProbabilistic Elastic–Plastic Response
Stochastic Elastic–Plastic Finite Element MethodSEPFEM FormulationsSEPFEM Verification Example
ApplicationsSeismic Wave Propagation Through Uncertain SoilsProbabilistic Analysis for Decision Making
Summary
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Seismic Wave Propagation Through Uncertain Soils
Applications
I Stochastic elastic–plastic simulations of soils andstructures
I Probabilistic inverse problems
I Geotechnical site characterization design
I Optimal material design
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Seismic Wave Propagation Through Uncertain Soils
Seismic Wave Propagation through Stochastic Soil
I Soil as 12.5 m deep 1–D soil column (von Mises Material)I Properties (including testing uncertainty) obtained through
random field modeling of CPT qT〈qT 〉 = 4.99 MPa; Var [qT ] = 25.67 MPa2;Cor. Length [qT ] = 0.61 m; Testing Error = 2.78 MPa2
I qT was transformed to obtain G: G/(1− ν) = 2.9qT
I Assumed transformation uncertainty = 5%〈G〉 = 11.57MPa; Var [G] = 142.32MPa2
Cor. Length [G] = 0.61m
I Input motions: modified 1938 Imperial Valley
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Seismic Wave Propagation Through Uncertain Soils
Random Field Parameters from Site DataI Maximum likelihood estimates
Typical CPT qT
Finite Scale
Fractal
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Seismic Wave Propagation Through Uncertain Soils
"Uniform" CPT Site Data
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Seismic Wave Propagation Through Uncertain Soils
Seismic Wave Propagation through Stochastic Soil
0.5 1 1.5 2Time HsL
-30
-20
-10
10
20
DisplacementHmmL
Mean± Standard DeviationJeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Probabilistic Analysis for Decision Making
OutlineMotivation
Stochastic Systems: Historical PerspectivesUncertainties in Material
Probabilistic Elasto–PlasticityPEP FormulationsProbabilistic Elastic–Plastic Response
Stochastic Elastic–Plastic Finite Element MethodSEPFEM FormulationsSEPFEM Verification Example
ApplicationsSeismic Wave Propagation Through Uncertain SoilsProbabilistic Analysis for Decision Making
Summary
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Probabilistic Analysis for Decision Making
Three Approaches to Modeling
I Do nothing about site characterization (rely onexperience): conservative guess of soil data,COV = 225%, correlation length = 12m.
I Do better than standard site characterization:COV = 103%, correlation length = 0.61m)
I Improve site characterization if probabilities of exceedanceare unacceptable!
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Probabilistic Analysis for Decision Making
Evolution of Mean ± SD for Guess Case
5
1015
20 25
−400
−200
200
400
600mean ± standard deviation
mean
Dis
plac
emen
t (m
m)
Time (sec)
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Probabilistic Analysis for Decision Making
Evolution of Mean ± SD for Real Data Case
5 1015
20 25
−200
−100
100
200
300
400
Time (sec)
Dis
plac
emen
t (m
m) mean ± standard deviation
mean
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Probabilistic Analysis for Decision Making
Full PDFs for Real Data Case
−400
0
200400
0
0.02
0.04
0.06
5
10
15
20
Tim
e (s
ec)
Displacement (mm)
−200
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Probabilistic Analysis for Decision Making
Example: PDF at 6 s
−1000 −500 500 1000
0.0005
0.001
0.0015
0.002
0.0025
Displacement (mm)
Real Soil Data
Conservative Guess
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Probabilistic Analysis for Decision Making
Example: CDF at 6 s
−1000 −500 500 1000
0.2
0.4
0.6
0.8
1
CD
F
Displacement (mm)
Real Soil DataConservative Guess
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Probabilistic Analysis for Decision Making
Probability of Exceedance of 20cm
5 10 15 20 25
10
20
30
40
50
60
70
Prob
abili
ty o
f E
xcee
danc
e of
20
cm (
%)
Displacement (cm)
With Conservative Guess
With Real Soil Data
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Probabilistic Analysis for Decision Making
Probability of Exceedance of 50cm
5 10 15 20 25
5
10
15
20
25
30
35
Displacement (cm)
Prob
abili
ty o
f E
xcee
danc
e of
50
cm (
%)
With Conservative Guess
With Real Soil Data
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Probabilistic Analysis for Decision Making
Probabilities of Exceedance vs. Displacements
10 20 30 40 50
20
40
60
80
Prob
abili
ty o
fE
xcee
danc
e (%
)
Displacemnent (cm)
Real Soil DataConservative Guess
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity
Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary
Summary
I Behavior of materials is probably probabilistic!
I Technical developments are available and are being refined
I Human nature: how much do you want to know aboutpotential problem?
Jeremic Computational Geomechanics Group
The Case for Probabilistic Elasto–Plasticity