seismic wave propagation in stochastic soils - …sokocalo.engr.ucdavis.edu/~jeremic/ · seismic...

Motivation Uncertain Soil Simulations Summary

Seismic Wave Propagation in Stochastic Soils

Boris Jeremic and Kallol Sett

Department of Civil and Environmental EngineeringUniversity of California,Davis, California, U.S.A.

4th ICEGE,Thessaloniki, Greece,

June, 2007

Jeremic, Sett Computational Geomechanics Group



Outline

MotivationSoils Behavior is Uncertain

Uncertain Soil SimulationsProbabilistic Soil Elasto–PlasticityStochastic Soil Dynamics

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)




Soils Behavior is Uncertain

Characterization and Quantification

I Natural variability of soil deposit (Phoon and Kulhawy1999, Fenton 1999) → function of soil formation process

I Ergodic assumption might not strictly apply (Der Kiureghian2005)

I Testing error (Phoon and Kulhawy 1999, Marosi andHiltunen 2004, 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.)




Probabilistic Soil Elasto–Plasticity

Our Original DevelopmentsConstitutive : Second-order accurate (exact mean and

variance) PDF of stress-strain response (Fokker –Planck – Kolmogorov Equation)

Spatial : Spectral Stochastic Elastic – Plastic FiniteElement Method, to simulate uncertain spatialvariability of elastic–plastic soils

I Obtain complete probabilistic description (PDF) for:I Stresses–Strain responseI Displacements (velocities, accelerations)

I Use for:I Sensitivity analysisI Probability of failure (tails of PDF)I Probabilistic site characterization design





Input Soil Parameters Random FieldsI Truncated Karhunen–Loevé (KL) expansionI Representation of input random fields in eigen-modes of

covariance kernel

Su(x , θ) = Su(x) +∑M

n=1√λnξn(θ)fn(x)∫

D C(x1, x2)f (x2)dx2 = λf (x1)

ξi(θ) =1√λi

∫D[Su(x , θ)− Su(x)]fi(x)dx

I Error minimizing propertyI Optimal expansion → minimization of number of stochastic

dimensions





Stochastic Elastic–Plastic Finite ElementsN∑

n=1

K e,epmn dni +

N∑n=1

P∑j=0

dnj

M∑k=1

CijkK′e,epmnk = 〈Fmψi [{ζr}]〉

Kmn =

∫D

BnDBmdV ; K ′mnk =

∫D

Bn√λkhkBmdV

Cijk =⟨ζk (θ)ψi [{ζr}]ψj [{ζr}]

⟩; Fm =

∫DφNmdV

I Based on SSFEM (Ghanem and Spanos (2003))I Random material and random forcingI Efficient representation of input random fields into finite

number of random variables using KL-expansionI Representation of (unknown) solution random variables

using polynomial chaos of (known) input random variables





Probabilistic Elasto–PlasticityI Probabilistic elastic–plastic constitutive incremental

equation ∆σij = De,epijkl ∆εkl

I Random stiffness De,epijkl

I Random strain increment ∆εkl

I Use of Euler Lagrange form of Fokker–Planck–Kolmogorov(FPK) equation (Kavvas 2003) to obtain

I Second-order accurate (exact mean and variance)stress–strain solution (PDF)

I Complete probabilistic description of response → PDF

I FPK equation is applicable to any elastic–plastic materialmodel





1–D Low and High OCR Cam Clay

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

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

random G, M and p0




Stochastic Soil Dynamics

1–D Shear Column ExampleI Static pushover of a

Stochasticshear column10m high (deep)

I Small correlationlength results inmean that tends todeterministic solution

I High correlationlength increasesmean andstandard deviation





Stochastic Seismic Ground Motions

−12

Prob

abili

ty D

ensi

ties

of D

ispl

acem

ent

Dis

plac

emen

t (m

)3

Time (s)

2 40

5 6 7 8

12

1

0

−2

8

4

0

6

10

5 10 15 20

−1

−0.5

0.5

1

Dis

plac

emen

t (m

)

Time (s)

5 10 15 20

0.05

0.1

0.15

0.2

0.25

0.3

Std.

Dev

. of

Dis

plac

emen

t (m

)

Time (s)

I Sinusoidal motions exampleI Complete PDF of motions

for each time step.I In general, increase in system uncertainty





Stochastic Seismic Ground Motions

1 2 3 4 5Time [s]

−0.002

−0.001

0.001

0.002

Displacement [m]

I Example motions: Imperial ValleyI Mean and SD of ground motionsI Large uncertainty at ground motion peaks




SummaryI A new, second-order accurate (exact mean and variance)

formulation for probabilistic elastic–plastic soil simulation

I Analytic modeling and simulations ofI spatial variability andI point-wise uncertainty

of soil elastic of elastic–plastic properties for static anddynamic problems

I Application to:I Sensitivity analysisI Probability of failure (tails of PDF)I Site characterization design (probabilistic)I General stochastic modeling in elastic–plastic solid and

structural mechanics



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