Unsaturated Soil Mechanics in Engineering€¦ · · 2017-12-01Why Did Unsaturated Soil Mechanics...

Why Did Unsaturated Soil Mechanics Not Emergence Along With Saturated Soil Mechanics?

Professor Emeritus Delwyn G. FredlundUniversity of Saskatchewan

Saskatoon, SK, Canada

The Tien H. Wu LectureOhio State University

Columbus, Ohio, U.S.A.May 12, 2006

The Tien H. Wu LectureDr. Tien H. Wu

Education & DegreesB.S. St. John's University, 1947, Shanghai, China M.S. University of Illinois, Champaign, IL, 1948 Ph.D. University of Illinois, Champaign, IL, 1951

Research and Teaching InterestsStrength properties of soil and rock Stability of embankments and natural slopes Groundwater and seepage Soil-reinforcement and soil-bio engineering Risk and reliability assessment for foundations and slopes

Tien H. Wu Lecture

Lifelong Dedication to Research and Teaching of Soil Mechanics

Tien H. Wu Lecture

Tien H. Wu Lecture

First Edition 1966

Second Edition 1976

3-D representation of the air-water interface (Contractile Skin)

Tortuosity


The Tien H. Wu LectureOhio State University

Columbus, Ohio, U.S.A.May 12, 2006


• Challenges (or Hindrances) to the development of Unsaturated Soil Mechanics

• Description of the Stress State• Fundamental Constitutive Relations• Role of the Soil-Water Characteristic Curve• Use of SWCC in the Constitutive Relations• Solution of a Series of PDEs• Modeling Unsaturated Soils Problems

{

Tien H. Wu Lecture

Objective

• To describe the Challenges Faced and the Solutions Generated in moving towards the Implementation of Unsaturated Soil Mechanics

Tien H. Wu Lecture

QUESTIONS TO KEEP IN MIND:

1.) Was it necessary for Unsaturated Soil Mechanics to lag behind Saturated Soil Mechanics?

2.) Will Unsaturated Soil Mechanics ever become routine in geotechnical engineering practice?

Gradual Emergence of Unsaturated Soil Mechanics

• 1950s: Independent measurement of pore-air and pore-water pressure through use of high air entry ceramic disks

• 1960s: Laboratory testing of unsaturated soils• 1970s: Constitutive relations proposed and tested

for uniqueness for unsaturated soils• 1980s: Solving formulations for classic Boundary

Value Problems• 1990s: Establishing procedures for determination of

unsaturated soil property functions• 2000+: Implementation into routine engineering

practice

Tien H. Wu Lecture

Challenges to the Implementation of Unsaturated Soil Mechanics

• Challenge #1:– To discover appropriate Stress State

Variables for describing the physical behavior of unsaturated soils

• Solution #1:

How have we done?How have we done?

Tien H. Wu Lecture

• Challenge #2: To develop devices that could measure a wide range of negative pore-water pressures (i.e., high matric suctions)

• Solution #2:



Tien H. Wu Lecture


• Challenge #3: – To develop (and test for uniqueness)

constitutive relations suitable for describing unsaturated soil behavior

• Solution #3:


Tien H. Wu Lecture


• Challenge #4: – To overcome the excessive costs

associated with the determination (i.e., measurement) of unsaturated soil properties (i.e., nonlinear functions)

• Solution #4:


Tien H. Wu Lecture


• Challenge #5: – To solve nonlinear partial differential

equations for unsaturated soils without having convergence difficulties during the iterative solution process

• Solution #5:


Tien H. Wu Lecture


• Challenge #6:– To promote and teach unsaturated soil

mechanics at universities and in engineering practice

• Solution #6:


Tien H. Wu Lecture

Regional distribution of unsaturated soils

Local vertical zones of

unsaturated soils

GROUNDWATER TABLE- Water filling the voids- Air in a dissolved state

SATU

RAT

ED

SOIL

Tien H. Wu Lecture

Zones of Desaturation Defined by a Soil-Water Characteristic Curve, SWCC

0

5

10

15

20

25

30

35

0.1 1.0 10. 100. 1000. 10,000 100,000 1000,000

Gra

vim

etric

wat

er c

onte

nt, w

(%)

Soil suction (kPa)

Air entry value

Boundary effect zone

Transition zone

Residual zone

Residualcondition

Inflection point

Tien H. Wu Lecture


• Challenge #1:– To discover appropriate Stress State

Variables for describing the physical behavior of unsaturated soils

• Solution #1:– Designation of independent Stress State Variables based on multiphase continuum mechanics principles

(σx - ua)

(σz - ua)

(σy - ua)(ua - uw)

(ua - uw)

τzx

τzy

τxz

τxy

τyzτyx

(ua - uw)

Tien H. Wu Lecture

Definition of stress state at a point in an unsaturated soil

(σz - ua)

(σy - ua)(ua - uw)

(ua - uw)

τzx

τzy

τxz

τxy

τyzτyx

(ua - uw)

(σx - ua)

• Defines the stress state at a point in a continuum

• State variables are independent of soil properties

Derivation of the Stress State is based on the superposition of equilibrium stress fields for a multiphase continuum

Tien H. Wu Lecture

State Variable Stage (Unsaturated Soils)

• Stress Tensors form the basis for a Science because we live in a 3-D Cartesian coordinate world

Net Total Stress Tensor

( )( )

( )

σ τ ττ σ ττ τ σ

x a yx zx

xy y a zy

xz yz z a

uu

u

−−

−

( )( )

( )

u uu u

u u

a w

a w

a w

−−

−

0 00 00 0

Matric Suction Stress Tensor

X - direction

Y - direction

Z - direction

Tien H. Wu Lecture

σ *ij = σij – [S uw + (1 – S) ua ] δ ij σ ij = total stress tensor, δij = Kroneker delta or substitution tensor,

σ *ij = Bishop’s soil skeleton stress (Jommi

2000) S = degree of saturation

Variations in Stress State Descriptionσ’ = (σ – ua) + χ (ua – uw)

σ’ = effective stress χ = parameter related to saturation

Above proposed equations are constitutive relations

Tien H. Wu Lecture

• Challenge #2:- To develop devices that can measure a wide range of negative pore-water pressures (i.e., high matric suctions)

• Solution #2:


- New instrumentation such as the high suction tensiometers and indirect thermal conductivity suction sensors provide viable techniques for the laboratory and the field

Tien H. Wu Lecture

Monitoring for Verification Purposes

• Measurement of matric suction:– Direct measurement techniques

• Low range tensiometers (< 90 kPa)• High range direct tensiometers (< 1200 kPa)

– Presently primarily for laboratory use– Indirect measurement techniques

• Thermal conductivity sensors• Measurements of water content: TDR Technology• Measurement of movement: same as for saturated

soils

Tien H. Wu Lecture

TDR ThetaProbe, ML2x manufactured by AT Delta Devices, U.K.

Monitoring of Water Content

Measures the dielectric constant for the soil around the rods. Dielectric constant varies with the water content of the soil

Tien H. Wu Lecture

Monitoring of Matric SuctionMeasures the thermal conductivity of a standard ceramic that varies in water content with the applied matric suction

Tien H. Wu Lecture

Matric Suction Versus Time - 1.0 m to 1.3 m Depth Range

0.0

25.0

50.0

75.0

100.0

125.0

150.0

175.0

200.0

15-Sep-00 5-Oct-00 25-Oct-00 14-Nov-00 4-Dec-00 24-Dec-00

Time (Days)

Mat

ric

Suct

ion

(kPa

)

T 1-3

T 2-8

T 3-11

T 4-14

T 5-16

T 4-14

T 1-3

T 2-8

T 3-11

T 5-16

In Situ Matric Suction measurements using Thermal Conductivity sensors at 1.0 to 1.3 m below roadway

Equalization

Frost

Time (Days)

Mat

ric s

uctio

n (k

Pa)

Tien H. Wu Lecture

Silicone rubber grommet

Rubber membrane

Latex rubber, to seal the rubbermembrane and grommet

Mini suction probe

O - ring5 – bar high air-entry ceramic disk

Direct, high suction sensor used to measure suctions greater than one atmosphere on the side of a triaxial specimen (Meilani, 2004)

Pore air pressure control

O - ringCoarse corundum disk

Filter paper

Specimen

Top cap

Water in the compartment is pre-pressurized to destroy cavitation nuclei

Tien H. Wu Lecture


• Challenge #3:– To develop (and test for uniqueness)

constitutive relations suitable for describing unsaturated soil behavior

• Solution #3:

Matric suction(ua - uw)

Voidratio

e

Net normal stress

(σ - u a)

am

at- Constitutive relations for saturated soils needed to be extended to embrace the effect of changing degrees of saturation

Tien H. Wu Lecture

Fundamental Constitutive Relations for Unsaturated Soils

• Constitutive Behaviors in Classic Soil Mechanics:– Seepage– Shear strength– Volume-mass changes: Void ratio, water content

changes

• Other topics in soil mechanics:– Heat flow (Freeze-Thaw and Evaporation)– Air flow– Contaminant transportEach constitutive relationship requires a nonlinear soil

property function; therefore, Unsaturated Soil Mechanics might be referred to as NONLINEAR SOIL MECHANICS!

Tien H. Wu Lecture

Water Seepage Constitutive Relations

Yg

uhw

w +=ρ

x wxdhv kdx

= −

dydhkv wyy −=

z wzdhv kdz

= −

Driving potential for water flow is hydraulic head, h

Darcy’s law (1856) for flow in the x-, y-, and z-direction

Coefficient of permeability, kw is a function of matric suction; therefore, the flow law is nonlinear and subject to hysteresis

Tien H. Wu Lecture

Shape of the water permeability function for glass beads tested by Mualem (1976 )

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

0.1 1 10 Soil suction (kPa)

Drying

Wetting

Coe

ffici

ent o

f per

mea

bilit

y (m

/s)

Wetting

Drying

DryingWetting

Soil suction (kPa)

Tien H. Wu Lecture

The SWCC for the glass beads showing hysteresis during drying and wetting

0

20

40

60

80

100

0.1 1 10 Soil suction (kPa)

Degr

ee o

f sat

urat

ion,

%

DryingWetting

Wetting

Drying

Soil suction (kPa)Hysteresis in the SWCC produces hysteresis in the Permeability function

Tien H. Wu Lecture

Water Storage in an Unsaturated Soil

0.1 1.0 10 100 1000 10000 100000 1E+6Soil suction, kPa

Wat

er c

onte

nt, %

0

10

20

30

40

Soil suction, kPa

Wat

er s

tora

ge

mod

ulus

, kPa

-1

0

2

4

8

0.1 1.0 10 100 1000 10000 100000 1E+6

Water storage function is the slope of the SWCC; Required for transient seepage analyses

Soil-Water Characteristic Curve, SWCC

Also has a hysteretic effect

Tien H. Wu Lecture

Air Flow Constitutive Relations

dxdukv a

axax −=

dydukv a

ayay −=

dzdukv a

azaz −=

Fick’s law for flow in the x-, y-, and z-direction

Coefficient of permeability, ka is a function of matric suction; therefore, the flow law is nonlinear and subject to hysteresis

Driving potential for air flow is Pore-air pressure, ua

Observation: Soil properties for unsaturated soils become nonlinear functions and are hysteretic in character

Tien H. Wu Lecture

Shear Strength Constitutive Relations

' '1( ) tan ( )n a a wc u u u fτ σ φ= + − + −

f1 = function showing the rate of increase in shear strength with matric suction

' '( ) tan ( ) tan bn a a wc u u uτ σ φ φ= + − + −

Linear form of the extended Mohr-Coulomb shear strength equation

Fredlund, Morgenstern and Widger, 1978

Tien H. Wu Lecture

Extended Mohr-Coulomb failure surface (Fredlund, Morgenstern and Widger, 1978)

φ’

(ua-uw)

Shea

r str

engt

h, τ

Net normal stress, (σ - ua)

Shear strength versus suction is nonlinear and affected by hysteresis

Air entry value

c’

Tien H. Wu Lecture

Multistage direct shear test results

on compacted glacial till (Gan et

al., 1988)

Soil-Water Characteristic Curve

for glacial till

0102030405060708090

100

1 10 100 1000 10000 100000 1000000Suction, (kPa)

Deg

ree

of S

atur

atio

n, S

(%)

0 kPa, e = 0.517

25 kPa, e =0.514

OPTIMUM INITIAL WATER CONTENT

SPECIMEN

AEV = 60 kPa

AEV = 60 kPa

Shea

r str

engt

h, τ

(kPa

)

200

250

150

100

50

00 100 200 300 400 500

c’ = 10 kPa(σf - ua)f tan φ’= 34.6 kPa

(σf - ua)f = 72.6 kPa φ’ = 25.5

Matric suction, (ua-uw) (kPa)

Tien H. Wu Lecture

Reference compression curves for a Saturated Soil

v =

(1+e

)

e

Log(σ)Ln(p)p0

λ

κ

Cs

σ0

Cc

Effective mean stress Effective vertical stress

Spec

ific

volu

me

Void

ratio

Preconsolidation pressure

Yield stress

ss CC 434.0

)10ln(≈=κ

cc CC 434.0

)10ln(≈=λ

Effective mean stress Effective vertical stress

Elasto-plastic form

Classic soil mechanics form

Tien H. Wu Lecture

Limiting or Bounding relationships for Ko

loading of a typical clayey silt soil

Residual

e

wGs

Air Entry value

106

Cc

Cs

e0

Cc

Cs

Air Entry value

Residual

Log (p-ua)

Log (soil suction)

Log (soil suction)

Log(p-u a)

Yield

Virgin compression line

p0-ua

Void ratio

Water content

Volume-mass constitutive relations for an unsaturated soil take the form of two, 3-dimensional surfaces with distinct breaks that identify changes in soil behaviorNew points:

- Air entry value- Residual point

Tien H. Wu Lecture

Volume–Mass Constitutive Surfaces

for Regina Clay – Preconsolidated at 200

kPa (Pham, 2004)

10000

1000

10010

1 0.10.01

Log net mean stress (kPa)

1e+061000001000010001001010.10.01Log soil suction (kPa)

0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

1.25

1.25

Deg

ree o

f satu

ratio

n (S

)

Deg

ree

of s

atur

atio

n (S

)

10000

1000100

101 0.1

0.01



0

0

0.5

0.5

1

1

1.5

1.5

2

2

2.5

2.5

Void

rat

io

Void

rat

io

10000

1000

10010

1 0.10.01



0

0

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.5

0.5

0.6

0.6

0.7

0.7

0.8

0.8

Gra

vim

etr

ic w

ate

r conte

nt

Gra

vim

etr

ic w

ate

r conte

nt

Basic volume-mass equation S e = w Gs

Void ratio, e Degree of saturation, S

Water content, w

SWCC

Tien H. Wu Lecture

Volume–Mass Constitutive Surfaces for Regina Clay Preconsolidated at 200 kPa (Pham, 2004)

Residual value

10000

1000100

101 0.1

0.01


1e+061000001000010001001010.10.01 Log soil suction (kPa)

0

0

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.5

0.5

0.6

0.6

0.7

0.7

0.8

0.8

G

ravi

met

ric w

ater

con

tent

G

ravi

met

ric w

ater

con

tent

Residual value

Air entry value

YieldYield

Water content, w

Soil suction Net total stress

SWCC

Tien H. Wu Lecture

Volume–Mass Constitutive Surfaces for

Beaver Creek Sand (Pham, 2004)

10000

1000100

101 0.1

0.01

Log net mean stress (kPa)1e+061000001000010001001010.10.01

Log soil suction (kPa)

0.3

0.3

0.34625

0.34625

0.3925

0.3925

0.43875

0.43875

0.485

0.485

0.53125

0.53125

0.5775

0.5775

0.62375

0.62375

0.67

0.67

Void

ratio

(e)

Void

ratio

(e)

10000

1000

10010

1 0.10.01



0

0

0.25

0.25

0.5

0.5

0.75

0.75

1

1

1.25

1.25

Degre

e o

f satu

ratio

n

Degre

e o

f satu

ratio

n

10000

1000

10010

1 0.10.01



0

0

0.05

0.05

0.1

0.1

0.15

0.15

0.2

0.2

0.25

0.25

0.3

0.3

Gra

vim

etr

ic w

ate

r conte

nt

Gra

vim

etr

ic w

ate

r conte

nt

Void ratio, e

Water content, w

Degree of saturation, S

SWCC

AEV

Basic volume-mass equation S e = w Gs

Tien H. Wu Lecture


• Challenge #4:– To overcome the excessive costs associated with the

determination (i.e., measurement) of unsaturated soil properties (i.e., nonlinear functions)

• Solution #4:

0.0

10

20

30

40

1E-1 1E+0 1E+1 1E+2 1E+3 1E+4Suction (kPa)

Predicted from grain-sizeExperimental

Wat

er c

onte

nt

1E+5 1E+6

- Indirect, estimation procedures have been developed to obtain unsaturated soil property functions based on Soil-Water Characteristic Curves

Tien H. Wu Lecture

Role and Measurement of the Soil-Water Characteristic Curve, SWCC

-Soil-Water Characteristic Curves, SWCC define the relationship between the amount of water in a soil and soil suction (i.e., matric suction and total suction)

- SWCC has been successfully used to estimate all unsaturated soil property functions

•ASTM Standard

•D6836-02 (2003)

0

4

8

12

16

20

0.1 1 10 100 1000Soil suction (kPa)

Gra

vim

etric

wat

er c

onte

nt (%

)Specimen 1Specimen 2Specimen 3

- Water permeability - Air permeability - Shear strength - Thermal flow - Incremental elasticity

Air Entry Value

Residual ValueSand

Tien H. Wu Lecture

0

5

10

15

20

25


Gra

vim

etri

c w

ater

con

tent

(%)

Specimen 1Specimen 2Specimen 3

AEV = 10 kPa

WEV = 4.5 kPa

Residual = 120 kPa

Residual = 62 kPa

Measured drying and wetting curves on processed silt (Pham, 2002)

Silt

Tien H. Wu Lecture

Pressure Plate Apparatus to Measure Void Ratio and Water Content While Applying Total Stress and Matric Suction

Manufactured by GCTS, Tempe, AZ

Air supply

High air entry disk

Tien H. Wu Lecture

Equations to Best-Fit SWCC Data

Fredlund and Xing (1994)

Correction FactorAir entry value

Rate of desaturation

Asymmetry Variable

ψ = Soil suction

fm

fn

f

s

ae

wCw

]})({ln[)()(

ψψψ

+×=

)1000000(1ln[

)1ln(1)(

r

rCψ

ψψ

ψ+

+−=

Numerous equations have been proposed:-Brooks & Corey (1964)- van Genuchten (1980)

0

4

8

12

16

20


Gra

vim

etri

c w

ater

co

nte

nt

(%)

Specimen 1Specimen 2Specimen 3

Wetting

Drying

ψψψ

Tien H. Wu Lecture

• Hysteretic SWCC Models will eventually be available for geotechnical usage

• Presently, the Geotechnical Engineer must decide which curve to use:– Select wetting curve or drying curve based on

process being simulated• Hysteresis loop shift at point of inflection:

– Sands: 0.15 to 0.35 Log cycle• Average: 0.25 Log cycle

– Loam soils: 0.35 to 0.60 Log cycle• Average: 0.50 Log cycle

Hysteresis in the Soil-Water Characteristic Curve

Estimation Values

Tien H. Wu Lecture

Model measurements of water content and matric suction showing the SWCC relationship from water contents and

matric suctions during wetting and drying simulations (Tami et al, 2004)

Section MSection B Section T

1 100.0

0.1

0.2

0.3

0.4

I&II-D

III-WIV&V-D

IV&V-W

III-D

1 10

I&II-D

III-WIV&V-D

IV&V-W

III-D

1 10

I&II-D

III-WIV&V-D

IV&V-W

III-D

Soil suction (kPa)

Volu

met

ric w

ater

con

tent

, θw

Suctions with Tensiometers Water content with TDR

Tien H. Wu Lecture

Approaches that can be used to obtain the Soil–Water Characteristic curves

Determination of Soil-Water Characteristic Curves, SWCC

Laboratory measurement of

water content versus suction

SWCC predictions from grain

size distribution

SWCC predictions

from grain size & Atterberg

limits

Dataset “mining” for

typical SWCC

Pressure plate <

1500 kPa

Vacuum desiccators > 1500 kPa

Numerous models

Parameters for

numerous models

Soils with similar grain size or soil

classification

Varying accuracy depending on the methodology used Tien H. Wu Lecture

Soil-Water Characteristic Curve computed from a Grain Size Distribution Curve

0.0

10

20

30

40

0.1 1 10 100 1000 10000Suction (kPa)

Predicted from grain-sizeExperimental

Wat

er c

onte

nt

1E+5 1E+6

020

40

60

80

100

0.0001 0.001 0.01 0.1 1 10 100Perc

ent p

assi

ng (%

)

Particle size (mm)

ExperimentalFit - curve

Fredlund et al,1997

Tien H. Wu Lecture

• Unsaturated soil property functions rely on the saturated soil properties PLUS the soil-water characteristic curve, SWCC

• SWCC (i.e., Air entry value and Residual value) can be incorporated through:– Direct insertion of the SWCC equation– Various forms of Integration along the SWCC– Differentiation of the SWCC

• Unsaturated soil property functions render the solution of a problem nonlinear

Incorporation of SWCC into the Constitutive Relations for Unsaturated Soils

Tien H. Wu Lecture

Seepage Constitutive Relations in Terms of SWCC

Mualem (1976)

Brooks & Cory (1964)

van Genuchten (1980)

Burdine (1953)

References for the Soil-Water Characteristic CurvePermeability Models

nn

mnn

rk 2

2

])(1[])(1[)(1)(

α ψα ψα ψψ

++−=

−−

5.0

21

])(1[}])(1[)(1{)( n

mnn

rk α ψα ψα ψψ

++−=

−−

λα ψψ 32)()( −−=rk

nm 21−=

nm 11−=

kr = kw/ksat

Tien H. Wu Lecture

Seepage Constitutive Relations in Terms of SWCC

Campbell (1974)Fredlund, Xing & Huang (1994)

Child and Collis –George (1950)

References for the Soil-Water Characteristic CurvePermeability Models

∫

∫−

−

= b

aev

yy

sy

by

y

y

r

dyeee

dyee

e

k

)ln(

'

)ln(

'

)()(

)()()(

ψ

ψ

θθθ

θψθθ

b

aevrk

22

)(−−

=ψ

ψ

b = Ln (1000000) θ (ψ) = Soil water content y = Dummy variable of integration representing

the logarithm of integration

kr = kw/ksat

Tien H. Wu Lecture

Usage of several functions to predict permeability functions from the SWCC for a particular soil and a suggested lower limit for the permeability function

1.E-171.E-161.E-151.E-141.E-131.E-121.E-111.E-101.E-091.E-081.E-071.E-061.E-051.E-04

0.1 1 10 100 1000 10000 100000 1E+06

Experimental data

Overall Kw + Kv

Fredlund, Xing & Huang

VaporKv

Campbell

Brooks and Corey

Van Genuchten - Mualem

van Genuchten - Burdine

Soil suction (kPa)

Coe

ffici

ent o

f per

mea

bilit

y (m

/s)

Tien H. Wu Lecture

Shear Strength Constitutive Equation Written in Terms of SWCC

Vanapalli et al. (1996)rs

rn θθ

θθ−

−=Θ

' ( ) tan ' ( ) tan 'a wn a nc u u uτ σ φ φ= + − + − Θ

Shear strength

Net normal stress on the failure plane

Angle of internal frictionEffective cohesion

Matric suction

SWCC

Tien H. Wu Lecture


• Challenge #5:– To solve nonlinear partial differential equations for

unsaturated soils without having convergence difficulties during the iterative solution process

• Solution #5:– Adaptive mesh (grid) generation techniques in

computer technology facilitates convergence

0 10 20 30 40 50 60 70 80 90 100 110 120 130 14001020304050

Tien H. Wu Lecture

• All classic areas of soil mechanics can be viewed in terms of the solution of a Partial Differential Equation

• Water flow through porous soils (Saturated or Unsaturated)

• Air flow through unsaturated soils• Stress analysis for slope stability, bearing capacity

and earth pressure• Stress-Deformation volume change and distortion

– Incremental elasticity– Elasto-plastic models

Problem Solving Environments, PSEs, for Soil Mechanics Partial Differential Equations, PDEs

Tien H. Wu Lecture

Solving a Boundary Value Problem

Element for which a Partial Differential Equation, PDE, must be derived

Boundary Value Must be Supplied

Boundary

Boundary

Boundary

Typical Boundary Conditions Flux Head - for seepage Force Displacement - for stress

Boundary

Utilize general purpose PDE Solvers to solve partial differential equations for saturated-unsaturated soil system

Must also define initial conditions

Tien H. Wu Lecture

Partial Differential Equation for Saturated-Unsaturated Water Flow Analysis

thm

yh

yk

yhk

xh

xk

xhk w

wwyw

y

wxw

x ∂∂γ−=

∂∂

∂∂

+∂∂+

∂∂

∂∂

+∂∂

22

2

2

2

Head variable to be solved

Water coefficient of permeability (function of soil suction)

Time Water storage (function of soil suction)

Tien H. Wu Lecture

Pore-air pressure(primary variable to be solved)

Air coefficient of permeability (function of soil suction)

Time Air storage and compressibility

(function of soil suction)

tu

RTgmuS

ee

yu

yk

xu

xk

yuk

xuk aaw

aaaaaaa

aa

a ∂∂

−

+−=

∂

∂∂

∂+

∂∂

∂∂+

∂∂+

∂∂ ω

22

2

2

2

1

Partial Differential Equation for Unsaturated Air Flow Analysis

Tien H. Wu Lecture

Partial Differential Equation for Saturated-Unsaturated Stress-Deformation Analysis

0441211 =

∂∂+

∂∂

∂∂+

∂∂+

∂∂

∂∂

xv

yuD

yyvD

xuD

x

44 12 11 0tu v u vD D D

x y x y x yγ

∂ ∂ ∂ ∂ ∂ ∂+ + + + = ∂ ∂ ∂ ∂ ∂ ∂ D11, D12, D44 = Combination of E and µ which are function of

soil suction and net total stresses

Stress-deformation analyses have a degrees of freedom in each of the Cartesian coordinate directions

X –

Y –

Tien H. Wu Lecture

• Convergence is the single most pressing problem facing modelers

• Most successful solutions have involved Adaptive Grid Refinement methods, AGR (Oden, 1989; Yeh, 2000)

• Mesh is dynamically upgraded during the solution based on error estimates

• AGR becomes extremely important when solving the nonlinear PDEs associated with Unsaturated Soil Mechanics

Convergence of Nonlinear Partial Differential Equations

Tien H. Wu Lecture

Two-dimensional seepage analysis through an earthfill dam with a clay core.

Equipotential lines

Optimized mesh for saturated-unsaturated seepage analysis

www.soilvision.com

Thieu and Fredlund, 1998

Tien H. Wu Lecture

Problem illustrating the solution of a 3-dimensional, saturated-unsaturated seepage PDEOptimized, automatically generated finite element mesh

Modeling of a waste tailings pond

Tien H. Wu Lecture

Stress analysis PDE combined with the Dynamic Programming procedure to compute the factor of safety

Distance0 20 40 60 80

0

5

10

15

20

25

30

DP Search Bounda ry

Fini te ElementS hear Stress

DP GeneratedCrit ical Slip SurfaceFOS= 1.3 Shape and location of the slip

surface are a part of the solution

Distance

Elev

atio

n

Pham and Fredlund, 2002 Tien H. Wu Lecture

• Requires the solution of a saturated-unsaturated seepage model and a stress-deformation model

Prediction of Heave or Collapse of a Soil

Saturated-Unsaturated Seepage Model

Saturated-Unsaturated Stress-Deformation

Model

Coupled Uncoupled

Pseudo-coupled

Computes changes in matric suction Computes deformations

Must consider effects of:-Nonlinearity-Coupling

Tien H. Wu Lecture

Consider Edge Lift for a Flexible Impervious Cover

Flexible cover 0

1

2

3

0 3 6 9 12

Distance from centre of cover or slab, m

CLConstant suction = 400 kPa

Flux = 0

Dep

th, m Flux = 0

Infiltration, q

Boundary conditions and initial conditions must be specified both seepage and stress-deformation

Tien H. Wu Lecture

Concrete slab0

1

2

3

0 3 6 9 12

CL

Dep

th, m

Distance from center, m

Can have one optimized Adaptive Mesh generated for seepage model and another for the stress-deformation model

Tien H. Wu Lecture

Matric Suction at Ground Surface after One Day of Infiltration for Various Infiltration Rates

0

100

200

400

0 2 4 6 8 10 12Distance from centre of cover, m

Mat

ric s

uctio

n, k

Pa

300

500

CL

Initial

q = 10 mm/day

q = 20q = 30

q = 40q = 50q = 60

Specifiedzero suction

Distance under slab

Tien H. Wu Lecture

Vertical Displacements at Ground Surface after One Day of Infiltration

0

10

20

25

0 2 4 6 8 10 12Distance from centre of cover, m

Hea

ve,(m

m)

5

15

specified zero suction

q = 10q = 20

q = 30q = 40

q = 50

q = 60 mm/day

CL

Distance under slab

Amount of edge heave

Tien H. Wu Lecture


• Challenge #6:– To promote implementation of unsaturated soil

mechanics into engineering practice• Solution #6:

- Educational materials and visualization tools have been produced to better teach and understand unsaturated soil mechanics

Tien H. Wu Lecture

Concluding Remarks

• Unsaturated Soil Mechanics needs to be first understood from the standpoint of the Constitutive equations describing soil behavior

• Constitutive Equations can be written in terms of the SWCC and are referred to as Unsaturated Soil Property Functions, USPF

• Direct and Indirect procedures are available for the assessment of the SWCC

• It is always possible to obtain an estimate of the required Unsaturated Soil Property Functions for geotechnical engineering applications

Tien H. Wu Lecture

Columbus, OhioMay 12, 2006

There are many soil mechanics researchers who deserve credit not only for researching saturated soil

behavior but also for improving our understanding of unsaturated soil behavior

Thank You

Tien H. Wu Lecture

Unsaturated Soil Mechanics in Engineering€¦ · · 2017-12-01Why Did Unsaturated Soil Mechanics...

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