SR Lecture 2 das

HD in Civil Engineering Subject (CSE352)

Soil and Rock Engineering岩土工程

Jian-Hua YIN 殷建華Office: TU731, Tel: 2766-6065Email: [email protected]

Outline of Lectures by JH YIN:Lecture 1: Subsoil Exploration (Chapter 2-Das)Lecture 2: Shallow Foundations (Chapter 3-Das)Lecture 3: Lateral Earth Pressure and Retaining Walls

(Chapter 5-Das)Lecture 4: Pile Foundations (Chapter 8-Das)Lecture 5: Stability of Slopes (Chapter 9-Craig) Lecture 6: Basic Rock Engineering (Chapters in Goodman 1989)Lecture 7: Soil Improvement and Ground Modification

(Chapter 8-Das plus others)Essential References:(1) Das, Braja M. (2007). Principles of Foundation Engineering (6th edition),

Thomson, United States (ISBN 0-534-40752-8)(2) Craig, R.F. (2004). Soil Mechanics, 7th edition (6thor 5th edition), Spon Press,

London and New York (ISBN 04-415-32702-2)(3) Goodman, R.E. (1989). Introduction to Rock Mechanics, 2nd edition, John

Wiley & Sons

3.1 IntroductionUltimate Bearing Capacity of Shallow Foundation3.2 General Concept3.3 Terzaghi’s Bearing Capacity Theory3.4 Factor of Safety3.5 Modification of Bearing Capacity Equations for

Water Table3.6 The General Bearing Capacity Equation3.7 Eccentrically Loaded Foundations (one way only)

Lecture 2: Shallow Foundations(Chapter 3-Das)淺基礎

Settlement of Shallow Foundation3.9 Types of Foundation Settlement3.10 Elastic Settlement Based on the Theory of

Elasticity3.11 Elastic Settlement of Foundations on Saturated

Clay3.13 Range of Material Parameters for Computing

Elastic Settlement


Primary Consolidation Settlement and Creep Settlement3.14 Primary Consolidation Settlement Relationships 3.15 Three-Dimensional Effect on Primary

Consolidation Settlement3.16 Vertical Stress Increase in a Soil Mass Caused by

Foundation Load (for Consolidation Settlement Calculation)

3.17 Allowable Bearing Pressure in Sand Based on Settlement Consideration

3.18 Field Load Test3.19 Presumptive Bearing Capacity3.20 Tolerable Settlement of Buildings


3.1 IntroductionA foundation(a) shall be safe against overall shear failure and (b) cannot undergo excessive displacement (or settlement)

Ultimate Bearing Capacity of Shallow Foundation

3.2 General Concept

Ultimate bearing capacity qu?Three failure modes?

)1.3(2*

LBBLB+

=

B = width of the foundation

L = length of foundation

A foundation has more settlement for punching shear failure than general shear failure

3.3 Terzaghi’s Bearing Capacity Theory

An equation for ultimate bearing capacity qu is derived considering force equilibrium

φφ

Ultimate bearing capacity qu at this level !!!

Terzaghi 太沙基 (1883-1963): 开創了土力學 – 土力學之父

)3.3(21'

γγBNqNNcq qcu ++=

c’ = cohesion of soilγ= unit weight of soilq = σ’v=γDf (equivalent surcharge) (no water) !Nc, Nq, Nγ= bearing capacity factors that are non-dimensional and are only functions of the soil friction angle, φ’Keep parameters (effective/total stress) consistent ! (保持一緻！)

For continuous foundation/strip footing:(considering one unit width, say 1 meter)

( ) )4.3(1cot1

24cos2

cot ''

2

tan)2/43(2'

''

−=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=−

qc NeN φφπ

φφφπ

)6.3(tan1cos2

1 ''2 φ

φγ

γ ⎟⎟⎠

⎞⎜⎜⎝

⎛−= pK

N

)5.3(

24cos2

'2

tan)2/43(2 ''

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=−

φπ

φφπeNq

Kpγ is the passive earth pressure coefficient

γγBNqNcNq qcu 4.03.1 ++= (square foundation) (3.7)

γγBNqNcNq qcu 3.03.1 ++= (circular foundation) (3.8)

'''

21

32

γγBNqNcNq qcu ++= (strip foundation) (3.9)

''' 4.0867.0 γγBNqNcNq qcu ++= (square foundation) (3.10)

''' 3.0867.0 γγBNqNcNq qcu ++= (circular foundation) (3.11)

For local shear failure - Use reduced (2/3) friction angle:

.2.3)'(tanusing,, 321'''' TableorcalculatedareNNN qc φφγ

−=

Terzaghi’s Bearing Capacity Theory Limitations:• No corrections on depth, load inclination, general foundation shape• Nγ is over-estimated.

3.4 Factor of Safety

Factor of Safety (FS) is defined in two ways:(a) Using gross ultimate bearing capacity qu:

)12.3(

)12.3(

bq

qFS

qqoraFSqq

design

u

alldesignu

all

=

≤=

The qall is gross allowable bearing capacity and qdesign is the design or current pressure on the foundation.

(3.12a) is for qall.(3.12b) is for assessment of the safety of the foundation!

FS is in the range 3 ~ 6

Factor of Safety (FS) is defined in two ways:(b) Using net (净) ultimate bearing capacity qnet(u):

fvdesignnetdesign

netdesign

u

netallnetdesignuunet

netall

uunet

Dqqqq

bq

qqFS

qqoraFS

qqFS

qq

qqq

γσ ==−=

−=

≤−

==

−=

;

)15.3(

)15.3(

)14.3(

)(

)(

)()()(

)(

)(

The qall(net) is net allowable bearing capacity and qdesign(net) is the net design or net current pressure/stress on the foundation.

(3.15a) is for qall(net).(3.15b) is for assessment of the safety of the foundation!Keep “net” consistent ! (保持一緻！)

Solution:From (3.7):From Table 3.1 for φ’=20o:

γγBNqNcNq qcu 4.03.1 ++=64.3,44.7,69.17 === γNNN qc

kNBBareaAqQloadgrossallowabletotalThe

mkNmkNFSqq

ThusmkN

q

allall

uall

u

5.292)5.15.1(130)(130)(:

/130/25.1304

85.520:

/85.52087.3843.13255.349

64.35.18.174.044.7)8.171(69.172.153.1

22

2

=××=××=×=

≈===

=++=

×××+××+××=

Example 3.2Repeat Example 3.1, assuming that local shear failure occurs in the soil supporting the foundations.

Solution:From (3.10):From Table 3.2 for φ’=20o:

kNBBareaAqQloadgrossallowabletotalThe

mkNFSqq

ThusmkN

q

allall

uall

u

4.133)5.15.1(3.59)(3.59)(:

/3.594

3.237:

/3.2370.121.692.156

12.15.18.174.088.3)8.171(85.112.15867.0

2

2

=××=××=×=

===

=++=

×××+××+××=

''' 4.0867.0 γγBNqNcNq qcu ++=12.1,88.3,85.11 ''' === γNNN qc

3.5 Modification of Bearing Capacity Equations for Water Table

Keep effective parameters consistent ! (保持一緻)

γsat = saturated unitweight of soil

γw = unit weight ofwater;

fDD ≤≤ 10

)( '' γγγγ −+=Bd

Case 1:

γγ BNqNNcq qcu''

21

++=

)16.3(

)('

21

21'

γγ

γγγσ

DD

DDq wsatv

+=

−+== 1D

Bd ≤≤0Case 2:

γγBNqNNcq qcu 21' ++=

)17.3(' γσ fv Dq ==

)18.3()( '' γγγγ −+=Bd

Bd >Case 3:

No change

γγBNqNNcq qcu 21' ++=

3.6 The General Bearing Capacity Equation (重要！)

Keep effective/total parameters consistent ! (保持一緻)

)19.3(21

idsqiqdqsqcicdcscu FFFBNFFFqNFFFcNq γγγγγ++=

c= c’ (effective) or c (total) cohesionγ = γ’ (effective), (average) or total unit weight γ of the soilB =width of foundation (=diameter of a circular foundation);

factorsninclinatioloadFFF

factorsdepthFFF

factorsshapeFFF

iqici

dqdcd

sqscs

=

=

=

γ

γ

γ

,,

,,

,,

γ

)22.3(tan)1(2)21.3(cot)1(

)20.3()2

45(tan tan2

φ

φ

φ

γ

φπ

+=

−=

+=

q

qc

q

NNNN

eN

(3.23) by Prandt; (1921), (3.22) by Reissner (1924), (3.24) by Caquot and Kerisel (1953), Vesis (1973). More equations for Nγ.

For effective stress parameters using φ’ and c’;

Total stress parameters using φ and c.

Keep consistent !

The three bearing capacity factors are calculated using the following 3 equations or Table 3.4:

Shape Factors: (by De Beer 1970)

widthBlengthfoundationLLBF

LBF

NN

LBF

s

qs

c

qcs

=>=

−=

+=

+=

)25.3(4.01

)24.3(tan1

)23.3(1

γ

φ

Depth Factors: (by Hansen 1970)

)28.3(1

)27.3()sin1(tan21

)26.3(4.01

1/

2

=

−+=

+=

≤

d

fqd

fcd

f

FB

DF

BD

F

BDFor

γ

φφ

Inclination Factors: (by Meyerhof 1963)

)!degree(tan

)31.3(1

)30.3(tan)sin1(tan21

)29.3(tan4.01

1/

1

12

1

notradiansinisB

DNoting

FB

DF

BD

F

BDFor

f

d

fqd

fcd

f

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

>

−

−

−

γ

φφ

verticalfromangleninclinatioload

F

FF

i

o

o

qici

=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−==

βφβ

β

γ )33.3(1

)32.3(90

1

2

2

β

Solution:

4.22,4.18304.3

/6.12187.0

:)21.3(,0

'

2

21

'

==>=

=×==

+=∴=

>

γ

γγγγ

φ

σ

γ

NNforTableFrom

mkNq

FFFBNFFFqNqfromc

waternotablewaterofmentionNo

qo

v

idsqiqdqsqu

Q

11.0302011;605.0

90201

901

1;202.017.0)30sin1(30tan21)sin1(tan21

:1/

6.04.01;577.1577.0130tan1tan1

2222

22

=⎟⎠⎞

⎜⎝⎛ −=⎟⎟

⎠

⎞⎜⎜⎝

⎛−==⎟⎟

⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

=+=−+=−+=

≤

=−==+=+=+=

φββ

φφ

φ

γ

γ

γ

io

o

o

o

qi

df

qd

f

sqs

FF

FBBB

DF

BDAssumeLBF

BB

LBF

0.13.1/7.0/3.1,&

43.489.1473.73150

150

43.489.1473.733/)3.1368.442.221(

3.1368.442.221

11.016.04.2218605.0)202.01(577.14.186.12

2

2

21

<==

++=

>==

++=++==

++=

××××××+×+××=

BDCheckmBfinderrortrialBy

BBB

BAreaQq

BB

BBFS

qq

Thus

BB

BB

q

f

allall

uall

u

3.7 Eccentrically Loaded Foundations (one way only)

0)2(3

4:6/

)61(6

)61(6:6/

)(

min

max

2min

2max

=−

=

≥

−=−=

+=+=

≤

=

qeBL

Qq

BeFor

Be

BLQ

LBM

BLQq

Be

BLQ

LBM

BLQq

BeFor

tyeccentriciQMe Q

Me =Q

Calculation of foundation base pressure for e<=B/6:

LBMpMBBLp iiii 2)()(

6;223

222

1=⇒=××××

)61()61(62)()( B

eBLQ

BQM

BLQ

LBM

BLQppp iii ±=±=±=±=

Q

H

e=

Q

H

M= +

QMe = )(i )(ii

BLQp i =)(

)(ip )(iip)(iip

6;0)61(min

BeBe

BLQp =⇒=−= Valid for e<=B/6 only

H H

Q

Calculation of foundation base pressure for e>B/6:

)&(21

max"

Qasmaglocationsamethe

QLqBF

FforceverticalispressuretheofareatotalThe

v

v

==

Q

vF"B

"

31

2BBe −=

"

31 B

)2(34

)323(

22

323,

31

2

"max

""

eBLQ

LeB

QLB

Qq

eBBBBe

−=

−==∴

−=∴−=Q

)43.3('21'

idsqiqdqsqcicdcscu FFFNBFFFqNFFFcNq γγγγγ++=

For one-way eccentricity1. Calculate effective width B’=B-2e, and effective length L’=L

Smaller of B’ and L’ is the effective width2. Use (3.43) to calculate qu

’. when calculate Fcs, Fqs and Fγs, use B’ and L’, but, Fcd, Fqd and Fγd, still use B and L (on safer side!).

3. The total ultimate load is:

4. The Factor of Safety (FS) is

5. The Factor of Safety (FS) may also be calculated using

)49.3()( ''''' LBqAqQ uuult ×==

)50.3(design

ult

QQFS =

max

'

qqFS u=

)43.3('21'

idsqiqdqsqcicdcscu FFFNBFFFqNFFFcNq γγγγγ++=

For one-way eccentricityWhy B’=B-2e?

Qe

eB 2−

eB−

2eB−

2

B

Settlement of Shallow Foundation3.9 Types of Foundation Settlement3.10 Elastic Settlement Based on the Theory of

Elasticity3.11 Elastic Settlement of Foundations on Saturated

Clay3.13 Range of Material Parameters for Computing

Elastic Settlement


3.9 Elastic Settlement Based on the Theory of Elasticity

Figure 3.14 Elastic settlement of flexible and rigid foundation

""""2/

?)5(40:''

1)(

'

'

2'

corneratsettlementforBBfoundationofcenteratsettlementforBB

BBzsoilofmodulussYoungAverageEsoilofratiosPoisson

pressureappliednetq

IIE

BqS

s

s

o

fss

soe

=

=

<<===

−=

μ

μα

Use Tables 3.4 and 3.5 to find Is and If

If foundation is perfectly flexible, the settlement is calculated as follows (from Bowles 1987):

),()( 93.0 centerflexibleerigide SS =

BHn

BLm

cornerAtBHn

BLm

centerAtsettlementforfoundationoflocationondependingfactora

===

===

=

''

''

,,1

:"")2/(

,,4

:"":

α

α

α

)(5

)()(

smallerwhicheverBorHz

averagez

zEE is

s

=

Δ= ∑

1

11)1(ln

)11()11(ln

)()(tan2

)(11

21)1934,(

2'2''

'

2

2'2''

2'2''

1

2'2''

2'2'2''

21

'

2

11

21

++=

+++

+++=

+++

+++=

=

+=

−−

+=

=

−

nmnmA

nmmnmmA

nmmnmmmA

radianinAnF

AAF

FF

erSteinbrennfactorshapeI

o

o

s

s

s

π

π

μμ

tableorfigurefromIFind

casesallinIDifNoting

BL

BD

f

FoxfactordepthI

f

ff

sf

f

1,0

),,(

)1948,(

==

=

=

μ

Example 3.6

Figure 3.16

use average Es

Figure 3.16

659.0031.0031

3.021641.0

121

21

=−

×−+=

−−

+= FFIs

ss μ

μ

659.0 mmm 2.120122.0 =

mmmm 39.112.12 =

Table 3.4 Eq. (3.69)

Figure 3.15

3.11 Elastic Settlement of Foundations on SaturatedClay

3.13 Range of Material Parameters for Computing Elastic Settlement

)(1: MPainNEKongHong s =

)(8.0 MPainNEs =

Primary Consolidation Settlement and Creep Settlement3.14 Primary Consolidation Settlement Relationships 3.15 Three-Dimensional Effect on Primary

Consolidation Settlement3.16 Vertical Stress Increase in a Soil Mass Caused by

Foundation Load (for Consolidation Settlement Calculation)

3.17 Allowable Bearing Pressure in Sand Based on Settlement Consideration

3.18 Field Load Test3.19 Presumptive Bearing Capacity3.20 Tolerable Settlement of Buildings


314 Primary Consolidation Settlement Relationships(One-Dimensional Straining – Vertical Compression Only)

'''

''''

'''

'

constant

0)(

)(21

)4(61

),,,,()(

)(;

1

mbt

bmtav

scoczz

zvz

o

oz

cz

ifH

zpc

CCfb

ma

einitialeratiovoideeestrainvertical

HdzSz

c

σσσ

σσσσ

σσσε

σε

ε

εεε

Δ≈Δ+Δ

≈Δ+Δ+Δ=Δ

=

Δ=

==

+Δ−

==

===

∫

'tσΔ

'mσΔ

'bσΔ

cHClay layer

conditionoedometerinsettlementS oedc −−

stresseffectiveverticialinitial

pressuredationpreconsoli

indexelasticswellingCC

indexncompressioCwhere

He

CHe

CS

withiiiAreasclayedconsolidatoverFor

He

CS

withiAreaclayedconsolidatoverFor

He

CHS

withiiAreaclayedconsolidatnormallyFor

o

c

es

c

cc

avcc

cspc

avco

cavs

pc

cav

cavc

czpc

cav

=

=

==

=

Δ++

++

=

Δ+<<

+−−

Δ++

=

<Δ+

−−

Δ++

==

≥Δ+

−

'

'

'

''0

0'0

'

0)(

''0

''

'0

''0

0)(

'''0

'0

''0

0)(

'''0

/

log1

log1

:)()(

log1

:)(

log1

:)(

σ

σ

σσσ

σσ

σσσσ

σσσ

σσσ

σσσε

σσσ

e'''zoz σσσ Δ+=

'cσ'

oσ'zσΔ

cC

es CorC

'zσ

)(i)(ii

conditionoedometerinsettlementS oedc −−

3.15 Three-Dimensional Effect on Primary Consolidation Settlement

22.3FigurefromratiosettlementK

conditionoedometerinsettlementS

KSS

oedc

oedcc

=

−=

−

−

22.3FigurefromratiosettlementKconditionoedometerinsettlementS

KSS

oedc

oedcc

=−

=

−

−

Stress due to a Concentrated LoadBoussinesg (1885) equation is

22

2/522 12

3:

yxrwhere

zrz

PincreasestressVertical

+=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+

=Δ

π

σ

3.16 Vertical Stress Increase in a Soil Mass Caused by Foundation Load (for Consolidation Settlement Calculation)

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+

−=Δ 2/320

21

11

:

zB

q

centrebelowincreasestressVertical

σ

Stress due to a Circularly Loaded Area

Stress below a Rectangular Area

8.3

,

n)f(m,factorinfluence)(2

)(3:

00 0 2/5222

30

TableUsezLn

zBm

I

Iqzyxzdxdyq

cornerthebelowincreasestressverticalThe

L

y

B

x

==

==

=++

=Δ ∫ ∫= = πσ

Table 3.8 Variation of Influence Value I

Table 3.8 Variation of Influence Value I

Below any point

say “O”

Figure 3.27

Calculate stress increase below the centre of a rectangular area

28.3.FigfromIIq

Iq

o

o

=Δ

=Δ

σ

σ

cCoe

For each layer Hj, if mv and Δσ’ are constant with depth z, then:

jvjvcj HmHS 'σε Δ==

In case of normally consolidated clay, using Cc:

jc

jvcj He

CHS '0

'1

0

log1 σ

σε+

==

For multi-layer Hj (j=1,2,3, …n), summation of settlements in all layers :called “分層縂和法”

∑=

=

=nj

jcjc ss

1

vU

ionconsolidatofdegreeaverageU

SUS

v

cvt

==

Settlement due to Secondary (Creep) Consolidation

)(1;2

log

2log

1 0)(

YinbydaytHtttC

Httt

eCHS

oco

o

co

oeczsc

=+

=

++

==

αε

αε

)/log(loglog 1212 tte

tteCC e

Δ−=

−Δ−

== αα

αεα Ce

C e =+ 01

For Hong Kong Marine Clays:

Cαe=0.3% to 1%

Why a clayey soil creeps?

Creep is due to –viscous adsorbed water (double layers) on clay particles–viscous re-arrangement/sliding/deformation of clay particles/plates–viscous deformation of clay plates

Adsorbed water is NOT free water which is free to flow under gravity.

Under effective stress

Creep movement !

• Creep always exists under the action of effective stresses (loading), independent of the excess pore water (or pore pressure).

• Therefore, creep has nothing to do with the“primary”consolidation.

• And creep exists during and after “primary”consolidation.

• Creep rate depends on stress/strain state:–Creep rate is large in a normally consolidated state.–Creep rate is small in a over-consolidated state.

Bjerrum’s time line model, apparent “pre-consolidation pressure”, ageing and “delayed compression” (Bjerrum 1967)

3.17 Allowable Bearing Pressure in Sand Based on Settlement ConsiderationMeyerhof (1956) proposed a correlation for the net allowable bearing pressure for foundation with SPT (N1)60 allowable settlement 25mm:

)22.1)((28.3

128.399.7)/(

)22.1)((98.11)/(2

602

)(

602

)(

)(

mmBforB

BNmkNq

mmeterinBforNmkNq

Dqq

allnet

allnet

fallallnet

>⎟⎠⎞

⎜⎝⎛ +

=

≤=

−= γ

Meyerhof and his wife in Newfoundland 1993 在加拿大：地基和桩基Photo was taken by JH Yin 1993 at an offshore platform construction site

3.17 Allowable Bearing Pressure in Sand Based on Settlement ConsiderationFor allowable settlement > 25mm (by Bowles 1977):

mminsettlementtolerableS

BDfactordepthF

mBforSFB

BNmkNq

mminBforSFNmkNq

e

fd

edallnet

edallnet

=

≤+==

>⎟⎠⎞

⎜⎝⎛ +

=

≤=

33.1)/(33.01

)22.1()25

(28.3

128.398.11)/(

)22.1)(()25

(16.19)/(

2

602

)(

602

)(

3.18 Field Load Test

platetestofcapacitybearingultimateqfundationproposedofcapacitybearingultimateq

whereqq

clayintestsFor

Pu

Fu

PuFu

=

=

=

)(

)(

)()(

:

platetestofwidthBfundationproposedofwidthB

whereBBqq

soilssandyintestsFor

P

F

P

FPuFu

==

= )()(

:

3.20 Tolerable Settlement of Buildings

ratiodeflectionL

EAlinereferencefromdeflectionrelative

lij

=Δ

−

=Δ

Δ==

=

)(

)jandipoints

betweendistanceis(

lS

distortionangular

pointssuccesibetwobetweengradient

''

ij

T(ij)β

α

In Hong Kong:(a)25mm – for important structures; (b) 50mm – less important(c) 100 mm for walk road, and (d) 200mm for gardens etc.

Professor A.W. Skemptonwas a well-respected and accomplished professor at Imperial College in the University of London有效應力和孔壓力係數

Problems:

For Figure 3.14, go to see Section 3.9 Elastic Settlement Based on the Theory of Elasticity

For Figure 3.27, go Section 3.16: Stress below a Rectangular Area

SR Lecture 2 das

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