enb371 - week3 -ln -elastic deformation of soil - 6 slides per page -colour

Elastic settlement in soilsElastic settlement in soils

ENB371: Geotechnical Engineering 2ENB371: Geotechnical Engineering 2Chaminda GallageChaminda Gallage

Chapter 5Chapter 5

BRAJA M. DASBRAJA M. DAS

Contents…Contents…

� Settlement in soils: Introduction

� Elastic settlement in sandy (granular) soils• Equation- based on theory of elasticity• Improved equation – based on theory of elasticity• Use of strain influence factor

� Elastic settlement in clay soils• Saturated clay (Janbu et al. 1956)

LowLowWeightWeight

WeakWeakSoilSoil

Large DistributedLarge DistributedWeightWeight

Weak RockWeak Rock

Very Large ConcentratedVery Large ConcentratedWeightWeight

Loading on the surface causes deformation/settlemen t in sub-soilsLoading on the surface causes deformation/settlemen t in sub-soils

Strong RockStrong Rock

Introduction (1)Introduction (1)

Introduction (2): Settlement in soilsIntroduction (2): Settlement in soils

structure embankment

Settlement profile

• These loads produce corresponding increases in the vertical effective stress, σσσσv’

• Settlement refers to the compression that soils und ergo as a response to the loads placing on the surface

Introduction (3): Significance Introduction (3): Significance

• If the settlement is not kept to tolerable limit, the desireduse of the structure may be impaired and the design life ofthe structure may be reduced

• It is therefore important to have a mean of predicting theamount of soil compression or settlement

Total settlement in soil due to loadingTotal settlement in soil due to loading

• Deformation of soil and rock grains

• Compression of air and water in voids

• Drainage of water and air from voids allowing compression of soil skeleton

• Creep movements –plastic adjustment of soil fabric under a constant effective stress

• Deformation of soil and rock grains

• Compression of air and water in voids

• Drainage of water and air from voids allowing compression of soil skeleton

• Creep movements –plastic adjustment of soil fabric under a constant effective stress

Elastic Deformation(Immediate

settlement) ( Se)

Elastic Deformation(Immediate

settlement) ( Se)

+ PrimaryConsolidation ( Sp)

+ PrimaryConsolidation ( Sp)

+ SecondaryCompression ( Ss)

+ SecondaryCompression ( Ss)

Se+Sp+Ss= Total SettlementSe+Sp+Ss= Total Settlement


TimingTiming

timetime

settl

emen

tse

ttlem

ent

log timelog time

elastic elastic -- immediate immediate -- fully recoverablefully recoverable

primary consolidationprimary consolidationdue to removal of waterdue to removal of waterinelasticinelastictimetime--dependentdependentpartial recovery onlypartial recovery only

1 week to several years1 week to several years

secondary compressionsecondary compressioncreep of particlescreep of particlesinelasticinelastictimetime--dependentdependentunrecoverableunrecoverable

0.5yr0.5yr 5yr5yr 50yr50yr


Elastic (immediate) settlement of soil ( Se)Elastic (immediate) settlement of soil ( Se)

AA

HH

FFFF

FFFF

uniformuniformaxialaxialstress, stress, σσσσσσσσ

∆∆∆∆∆∆∆∆

∆∆∆∆∆∆∆∆ = = σσσσσσσσ HH / / EE

σσσσσσσσ = F = F // AA

ε = ∆/Ηε = ∆/Ηε = ∆/Ηε = ∆/Ηε = ∆/Ηε = ∆/Ηε = ∆/Ηε = ∆/Η

σ = εσ = εσ = εσ = εσ = εσ = εσ = εσ = εEE


•Caused by the elastic deformation of soil particles when the effective stress is increased•Fully recoverable•Not time dependent (occur within very short time)•Calculations generally based on equations derived f rom theory of elasticity

For

σ

∆orε

For

σ

∆orε

Elastic properties ( E – Young’s modulus) of clay and sand

Elastic properties ( E – Young’s modulus) of clay and sand

In the case of an extensive, homogeneous deposit ofsaturated clay, it is a reasonable approximation to assumethat E is constant throughout the deposit

In the case of sand, however, the value of E varies withconfining pressure and, therefore, will increase with dept hand varies across the width of the loaded area


Flexible vs Rigid FoundationFlexible vs Rigid Foundation

Flexible foundation : uniform contact pressure, non-uniform deformationRigid foundation : non-uniform contact pressure, uniform deformation


Flexible vs Rigid Foundation(contact pressure and settlement )

Flexible vs Rigid Foundation(contact pressure and settlement )

(a) Flexible foundation

(b) Rigid foundation


Contours of equal vertical pressure in the vicinity of (a) a strip area carrying a uniform pressure(b) a square area carrying a uniform pressure

Contours of equal vertical pressure in the vicinity of (a) a strip area carrying a uniform pressure(b) a square area carrying a uniform pressure

The zone lying inside the vertical stress contour of value 0.2q is described as the bulb of pressure

B – width of the foundation

Elastic settlement of sandy soil under flexible fou ndation (Equation based on theory of elasticity ) - 1


fss

se II

EBqS

2

0

1)'(

µα −=

Bowels (1987)Bowels (1987)

)1948,(

)1934,(

2/'

40

from elasticity of modulus Average

'0

FoxfactordepthI

erSteinbrennfactorshapeI

foundationofcornerforB

foundationofcentretheforBB

Bzabouttoz

E

soilofratiosPoisson

foundationtheonpressureappliednetq

f

s

s

s

====

=====

µ

fundationofcornertheat

foundationofcentretheat

,1

,4

==

αα

level foundation fromrock theDepth to

foundation theofDepth D

foundation theofLength L

foundation theofWidth

f

==

==

H

B

Is= shape factorIs= shape factor

21 1

21FFI

s

ss µ

µ−

−+=

F1 and F2 can be obtained from tables using m’ and n’ or can be calculated

F1 and F2 can be obtained from tables using m’ and n’ or can be calculated

)2/(''

B

Hn

B

Lm

foundationtheofcentretheat

settlementthecalculateTo

==B

Hn

B

Lm

foundationtheofcorneraat


== ''



fss

se II

EBqS

2

0

1)'(

µα −=

Variation of F 1with m’ and n’Variation of F 1with m’ and n’

Variation of F 2with m’ and n’Variation of F 2with m’ and n’

Is( shape factor ) can be calculatedIs( shape factor ) can be calculated21 1

21FFI

s

ss µ

µ−

−+=

21

2

101

tan2

'

)(1

An

F

AAF

−=

+=

π

π

)1'''

'1'''

'1)1''(ln

)1''1('

'')1'1(ln'

222

22

22

1

22

222

0

++=

++++++=

++++++=

nmn

mA

nmm

nmmA

nmm

nmmmA



)2/(''

B

Hn

B

Lm

foundationtheofcentretheat


==B

Hn

B

Lm

foundationtheofcorneraat


== ''

If= depth factorIf= depth factor

=

B

L

B

DfI s

ff ,,µ

If can be obtained from charts using D f/B, µµµµs and L/B If can be obtained from charts using D f/B, µµµµs and L/B



fss

se II

EBqS

2

0

1)'(

µα −=



Due to the heterogeneous nature of soil deposits, t he magnitude of E s may vary with depth. For that reason, Bowles (1987) recommended using average of E s

Due to the heterogeneous nature of soil deposits, t he magnitude of E s may vary with depth. For that reason, Bowles (1987) recommended using average of E s

z

zEE is

s∑ ∆

= )(

smalleriswhicheverBorHz

zdepthawithinelasticityofulussoilE is

,5

mod)(

=

∆=

fss

se II

EBqS

2

0

1)'(

µα −=



Example 1 - 1Example 1 - 1A flexible shallow foundation 1 m X 2 m is shown be low. Calculate the elastic settlement at the centre of t he foundation.

A flexible shallow foundation 1 m X 2 m is shown be low. Calculate the elastic settlement at the centre of t he foundation.

fss

se II

EBqS

2

0

1)'(

µα −=

Example 1 - 2Example 1 - 2

fss

se II

EBqS

2

0

1)'(

µα −=

z

zEE is

s∑ ∆

= )(

B = 1 m, L = 2 mB = 1 m, L = 2 m Bmz 55 ==

kPaEs 400,105

)2000,12()1000,8()2000,10( =×+×+×=

For centre of foundationFor centre of foundation

105.0

5

)2/(',2

1

2',4 =======

B

Hn

B

Lmα 5.0

2

1

2' === B

B

Example 1 - 3Example 1 - 3

mmIIE

BqS fss

se 27.1201227.0709.0659.0

400,10

)3.01(5.04150

1)'(

22

0 ==××−×××=−= µα

10',2' == nm From tables, F 1 = 0.641 F2 = 0.031From tables, F 1 = 0.641 F2 = 0.031

659.0031.03.01

3.021641.0

1

2121 =×

−×−+=

−−+= FFI

s

ss µ

µ

709.0,3.0,21

2,1

1

1 ====== fSf Ifiguresfrom

B

L

B

Dµ

Improved equation for Elastic Settlement of sandy soil-1


Mayne and Poulos (1999) presented an improved formu la for calculating the elastic settlement of soil (sand) u nder foundation load taking into account:

•Rigidity of the foundation

•Depth of embedment of the foundation

•Increase in elastic modulus of the soil with depth

•Location of rigid layers at limited depth

Mayne and Poulos (1999) presented an improved formu la for calculating the elastic settlement of soil (sand) u nder foundation load taking into account:

•Rigidity of the foundation

•Depth of embedment of the foundation

•Increase in elastic modulus of the soil with depth

•Location of rigid layers at limited depth

( )2

0

0 1 sEFGe

e E

IIIBqS µ−=

foundationofthicknesst

factorcorrectionembedmentFoundationI

factorcorrectionrigidityFoundationI

depthwithEof

iationtheforfactorInfluenceI

depthFoundationD

soilratiosPoisson

foundationofulusElasticE

layersoillecompressibofulusElasticE

foundationofdiameterEquivalentB

foundationof

centrethebelowsettlementelasticTheS

E

F

s

G

f

s

f

S

e

e

===

=

==

===

=

var

'

mod

mod

µ



level foundation fromrock theDepth to=H

For rectangular foundationFor rectangular foundation

foundationoflengthL

foundationofwidthB

BLBe

==

=π

4

For circular foundationFor circular foundation

foundationofdiameterB

BBe

==

kzEEs += 0

zwithuluselasticsoilinincreaseofratek

foundationofbottomthefromdepthz

depthfoundationatsoilofelasticityE

mod

0

==

=



( )2

0

0 1 sEFGe

e E

IIIBqS µ−=

Influence factor for the variation of E s with depthInfluence factor for the variation of E s with depth

),( 0

eeG B

H

kB

EfI == β

Improved equation for Elastic Settlement of sandy soil- 4

Improved equation for Elastic Settlement of sandy soil- 4 ( )2

0

0 1 sEFGe

e E

IIIBqS µ−=

Foundation rigidity correction factor

Foundation rigidity correction factor

3

0

2

2

106.4

1

4

++

+=

ee

f

F

B

t

kB

E

E

Iπ


Improved equation for Elastic Settlement of sandy soil- 5 ( )2

0

0 1 sEFGe

e E

IIIBqS µ−=

Foundation embedment correction factorFoundation embedment correction factor

( )

+−

−=6.14.022.1exp5.3

11

f

e

E

D

BI

µ

( )2

0

0 1 sEFGe

e E

IIIBqS µ−=



Elastic settlement of sandy soil using strain influence factor - 1


The settlement of granular soils can also be evalua ted by the use of semiemperical strain influence factor propos ed by Schmertmann (1978).

The settlement of granular soils can also be evalua ted by the use of semiemperical strain influence factor propos ed by Schmertmann (1978).

zE

IqqCCS

z

s

ze ∆−= ∑

2

021 )(

level foundation thefromincrement Depthez

soil of elasticity of Modulus

factor influenceStrain

2

1

=∆=

=====

s

f

z

E

Dq

foundationtheofleveltheatstressq

soilincreepforaccounttofactorcorrectionaC

embedmentfoundationofdepththeforfactorcorrectionaC

I

γ

+=

−−=

1.0log2.01

5.01

2

1

yearsintimeC

qq

qC



zE

IqqCCS

z

s

ze ∆−= ∑

2

021 )(

Correction factor for foundation depth (C 1) and correction factor for creep (C2) are given by:

Correction factor for foundation depth (C 1) and correction factor for creep (C2) are given by:

The strain influence factor I z for square (L/B=1) or circular foundations and foundations with L/B ≥10 is shown here. I z diagrams for foundations with 1<L/B<10 can be linearly interpolated

The strain influence factor I z for square (L/B=1) or circular foundations and foundations with L/B ≥10 is shown here. I z diagrams for foundations with 1<L/B<10 can be linearly interpolated



zE

IqqCCS

z

s

ze ∆−= ∑

2

021 )(

The maximum value of strain influence factor , I z(m) can be calculated as:The maximum value of strain influence factor , I z(m) can be calculated as:



)1()( '

1.05.0z

mz q

qqI

−+=

level foundation thefrom measured is z

foundation theofon constructi before z ofdepth aat stress Effective' 1)1( =zq



Procedure for calculation of S e using the strain influence factorProcedure for calculation of S e using the strain influence factor

Step1: Plot the variation of I z with depth

Step2: Plot the actual variation of E s of soil with depth. E s can be calculated from SPT or CPT results

Step3: Approximate the actual variation of E s in to number of layers of soil having constant E s

Step4: Divide the soil layer from z=0 to z=z2 into number of layers which will depend on breaking in continuity in I zand E s diagrams

Step5: Prepare a table to obtain

Step6: Calculate C 1 and C 2

Step1: Plot the variation of I z with depth

Step2: Plot the actual variation of E s of soil with depth. E s can be calculated from SPT or CPT results

Step3: Approximate the actual variation of E s in to number of layers of soil having constant E s

Step4: Divide the soil layer from z=0 to z=z2 into number of layers which will depend on breaking in continuity in I zand E s diagrams

Step5: Prepare a table to obtain

Step6: Calculate C 1 and C 2

zE

I

s

z ∆∑

Example 2 -1Example 2 -1

A 3 m wide strip foundation on a deposit of sand layer is shown along with the variation of modulus of elasticity of the soil (E s). The unit weight of sand is 18 kN/m 3.

Calculate the elastic settlement of foundation after 10 years using the strain influence factor.

A 3 m wide strip foundation on a deposit of sand layer is shown along with the variation of modulus of elasticity of the soil (E s). The unit weight of sand is 18 kN/m 3.

Calculate the elastic settlement of foundation after 10 years using the strain influence factor.

1.5 m

kPaq 2000 =

)(MPaEs

3/18 mkN=γ

zE

IqqCCS

z

s

ze ∆−= ∑

2

021 )(

Example 2 -2: Distribution of strain influence factor, Iz

Example 2 -2: Distribution of strain influence factor, Iz

100.3, ≥∴==∝B

LBL

9/)3(*5.05.0 −−= zI z

kPaqq 2000 ==

kPaDq

mDmkn

f

f

275.118

,5.1,/18 3

=×==∴

==

γγ

)1()( '

1.05.0z

mz q

qqI

−+=

kPaq z 81)35.1(18' )1( =+×=

65.0)( =mzI

81

272001.05.0)(

−+=mzI

Example 2 -3Example 2 -3 100.3, ≥∴==∝B

LBL

kPaq 2000 =

)(MPaEs

3/18 mkN=γ

2.0=zI

5.0=zIzI

z

z 1.02.0

30

+=≤≤

)3(9/5.05.0

123

−−=≤≤

zI

z

z 9/)3(*65.065.0 −−= zI z

65.0)( =mzIzI mz 15.02.0)( +=

z

)0.3( mz =

)0.12( mz =

zE

IqqCCS

z

s

ze ∆−= ∑

2

021 )(

Example 3 -4Example 3 -4Layer No ∆∆∆∆z [m] Es

[kPa]Z to the middle of layer

Iz at the middle of layer

1 2.0 6000 1.0 0.35 0.000117

2 1.0 12000 2.5 0.575 0.0000479

3 4.5 12000 5.25 0.4875 0.000183

4 4.5 10000 9.75 0.1625 0.0000731

]/[ 3 kNmzE

I

s

z ∆

kNmkNmzE

I

s

z /000421.0]/[ 33 =∆∑

Example 3 -4Example 3 -4

mmmzE

IqqCCS

z

s

ze 94094.0000421.0)27200(4.1922.0)(

2

021 ==×−××=∆−= ∑

4.11.0

10log2.01

1.0log2.01

922.027200

275.015.01

2

1

=

+=

+=

=−

×−=

−−=

yearsintimeC

qq

qC

kPaDq

kPaq

f 275.118

200

=×===

γ

kNmkNmzE

I

s

z /000421.0]/[ 33 =∆∑

Elastic settlement of foundation on Saturated Clay

Elastic settlement of foundation on Saturated Clay

Janbu (1956) proposed an equation for evaluating th e average settlement of flexible foundation on saturated clay soils Janbu (1956) proposed an equation for evaluating th e average settlement of flexible foundation on saturated clay soils

se E

BqAAS 0

21=

A1 is a function of H/B and L/BA2 is a function of D f/BA1 is a function of H/B and L/BA2 is a function of D f/B

foundation on the pressure appliedNet q

clay of elasticity of ModulusE

level foundation fromrock theDepth to

foundation theofDepth D

foundation theofLength L

foundation theofWidth

0

s

f

====

==

H

B

Elastic settlement of sandy soil under rigid foundation

Elastic settlement of sandy soil under rigid foundation

),()( 93.0 centreflexibleerigide SS ≈

LoadLoad

stressstress

deflectiondeflection

LoadLoad

deflectiondeflection

stressstress

Elastic Modulus, E (MPa)Elastic Modulus, E (MPa)

1,00

01,

000

200,

000

200,

000

100,

000

100,

000

10,0

0010

,000100

1001010

11

Ste

elS

teel

(210

k)(2

10k)

Con

cret

eC

oncr

ete

(20k

(20k

--40k

)40

k)

Har

dwoo

dH

ardw

ood

(5k

(5k

--20

k)20

k)

SoilSoil RockRock

Har

d C

lay

Har

d C

lay

Sof

t Cla

yS

oft C

lay

Den

se S

and

Den

se S

and

Loos

e S

and

Loos

e S

and

))

Gra

nite

Gra

nite

(50k

(50k

--100

k)10

0k)

Mud

ston

eM

udst

one

(200

(200

--8k)8k)

San

dsto

neS

ands

tone

(3k

(3k--

20k)

20k)

XWXWHWHW

MWMWSWSW

FFCWCW

Mud

ston

eM

udst

one

(200

(200

--8k)8k)

Elastic parameters Elastic parameters -- clayclayElastic parameters Elastic parameters -- clayclay

EE• Soft clay• Medium clay• Stiff Clay

νννννννν• All saturated clays

• 4.1 – 20.7 MPa• 20.7 – 41.4 MPa• 41.1 – 96.6 MPa

• 0.5 (no vol. change)

Elastic parameters Elastic parameters -- sandsandElastic parameters Elastic parameters -- sandsand

EE•• Loose sandLoose sand•• Medium sandMedium sand•• Dense sandDense sand•• Silty sandSilty sand•• Sand and gravelSand and gravel

νννννννν•• Loose sandLoose sand•• Dense sandDense sand

•• 10 10 -- 24 MPa24 MPa•• 17 17 -- 28 MPa28 MPa•• 34 34 -- 55 MPa55 MPa•• 10 10 –– 17.25 MPa17.25 MPa•• 69 69 –– 172 Mpa172 Mpa

•• 0.1 to 0.30.1 to 0.3•• 0.3 to 0.40.3 to 0.4

SummarySummary

How to calculate elastic settlement in sandy (granular) soils?

• Equation- based on theory of elasticity• Improved equation – based on theory of elasticity• Use of strain influence factor

How to calculate elastic settlement in clay soils• Saturated clay (Janbu et al. 1956)

enb371 - week3 -ln -elastic deformation of soil - 6 slides per page -colour

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