10) Foundation SettlementsBased on the theory of elasticity, the elastic settlement of a shallow foundation S e can be defined as, =∫ = ∫∆ − ∆ − ∆ H z s x s y s H Se

Geotechnical Engineering

SNU Geotechnical and Geoenvironmental Engineering Lab.

37

10) Foundation Settlements

� Immediate (Elastic) settlement � Both clays and sands

� Time dependent settlement (clays) �

�

(Primary) consolidation settlement

Secondary compression settlements

(significant in highly plastic and

organic soils)

- An example of time-displacement curve from consolidation tests of clays.

Primary consolidation Secondary compression

time (log scale) t100

dis

pla

ce

men

t



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i) Immediate Settlements

� Settlements due to elastic deformations of soil mass.

� Based on the theory of elasticity, the elastic settlement of a shallow foundation Se can be defined as,

∫∫ ∆−∆−∆==H

ysxsz

s

H

ze dzE

dzS00

)(1

σµσµσε

where, sE : modulus of elasticity of soil

sµ : Poisson’s ratio of soil



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� Theoretically, for 0=fD , ∞=H and perfectly flexible foundations,

')1( 20 αµ s

s

eE

BqS −= (Harr, 1966)

where, 'α is a is a function of shape and flexibility of foundation

and location of concerning points.



40

flexible foundation : at center, αα ='

at corner, 2/' αα =

average, aveαα ='

rigid foundation : rαα ='

� Comments

Harr’s equation generally results in too conservative value

(i.e. overestimating settlement).

a) H (depth to the relatively incompressible layer) < 2B ~ 3B eS

b) The deeper the embedment fD , the lesser is eS .



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� Es may vary with depth.

� Bowles (1987) recommended to use a weighted average of Es,

z

zEE

iiss

∆Σ=

)(

∆Z1

∆Z2

∆Z3

∆Z4 Es(4)

Es(3)

Es(2)

Es(1)

H

Es



42

� Estimation of elastic settlements by Mayne and Poulous (1999),

- It takes into account

1) the rigidity of the foundation

2) the depth of embedment of the foundation

3) the increase in the modulus of elasticity of the soil with depth

4) the location of rigid layers at a limited depth

)1( 2

0

0s

EFGee

E

IIIBqS µ−=

where, eB : the equivalent diameter

= πBL4

for rectangular foundation

= B for circular foundation



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GI : influence factor for the variation of )( 0 kzEEs += with depth

=

=

ee B

H

kB

Ef ),( 0β � Fig. 5.17

FI : foundation rigidity correction factor

= 3

0

2

2

106.4

1

4

++

+

ee

f

B

t

kB

E

E

π � Fig. 5.18

EI : foundation embedment correction factor

=

)6.1)(4.022.1exp(5.3

11

+−−

f

es

D

Bµ

� Fig. 5.19

<Figure 5.17 Variation of GI with β >



44

<Figure 5.18 Variation of rigidity correction factor FI with flexibility factor FK >

<Figure 5.19 embedment correction factor EI with eF BD / >

I E



45

� Immediate settlement of sandy soil : use of strain influence factor

(Schmertmann and Hartman (1978))

∑ ∆−=z

s

ze z

E

IqqCCS

0

21 )(

where :

1C = A correction factor for the depth of foundation embedment

( )]/([5.01 qqq −−= )

2C = A correction factor for creep

= 1.0

log2.01t

+ � t in years

zI = Strain influence factor (chiefly related to shear stress increase

in soil mass due to foundation load)



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For square, circular ft. For strip footing (L/B >10)

0=z 1.0=zI 0=z 2.0=zI

Bz 5.01 = 5.0=zI Bz =1 5.0=zI

Bz 22 = 0=zI Bz 42 = 0=zI

For rectangular ft, interpolate two cases.

- This method is effective for layered soils (E is varying with depth.)



47

� Determination of Young’s modulus

- sE and sµ can be determined from the laboratory tests.

- Correlation between sE and SPT and CPT results.

Schmertmann(1970) ; for sands

608Np

E

a

s =

where 60N : standard penetration resistance

ap : atmosphere pressure kPa100≈

cs qE 2=

where cq : static cone resistance

Schmertmann and Hartman(1978) ; for strain influence factors,

cs qE 5.2= for square and circular foundation

cs qE 5.3= for strip foundation

sE for clays

uus ctocE 500250= for normally consolidation clays

uus ctocE 1000750= for overconsolidated clays



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ii) Time-dependent settlements

Primary consolidation �

Secondary compression �

� Consolidation settlement

∫= dzS vc ε ( vε : vertical strain)

∫ +∆

= dze

e

01

'pσ : Maximum past pressure

cC : Compression index

sC : Swelling index

'0σ : Average present effective stress 'aveσ∆ : Average increase of pressure on the clay layer by loading

e0

Cs

∆e

e

'pσ (log scale)

Cc

'pσ

'0σ

''0 aveσσ ∆+



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- When thickness of clay layer is cH and initial void ratio is e0,

)'

''log(

1'log

1

0

0

'

0 p

ave

p

ccpsc

ce

CH

e

CHS

σ

σ∆+σ

++

σ

σ

+=

For normally consolidated clay ( p''0 σσ = ),

)'

''log(

1 0

0

0 σσσ avecc

ce

CHS

∆+

+=

For overconsolidated clay ( p''0 σσ < ),

a) )'

''log(

1'''

0

0

0

0 σσσ

σσσ avesccpave

e

CHS

∆+

+=≤∆+

b) )''

log(1

)'

log(1

''''

0

00

'

0

0

p

aveccpsccpave

e

CH

e

CHS

σσσ

σ

σσσσ

∆+

++

+=>∆+



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- How to determine avσ∆ and 0σ

i) )4(6/1 bmtav σσσσ ∆+∆+∆=∆

0σ = effective stress at the middle of clay layer.

⇒ Calculate cS

ii) Divide the clay layer into several thin layers

⇒ σ∆ and '0σ are obtained from the middle of each thin layers.

⇒ Calculate settlements of each thin layers, ciS .

⇒ Total consolidation settlement, ∑= cic SS



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� Estimation of stress increase on the clay layer due to external loads

(Uniform load and flexible foundation)

- σ∆ determined from Bousinesq approach

(Assuming soil as a semi-infinite, elasitc, isotopic, and

homogeneous medium.)

m = B/z, n=L/z

a) The stress increase σ∆ below the corner of rectangular loaded area.

Iq0=∆σ

where values of I are given in Table 5.2.

(or σ∆ can be directly calculated with Eq (5.5) and (5.6) in

textbook.)



52

b) Stress increase below any point below rectangular loaded area.

)( 4321 IIIIqo +++=∆σ

- Approximate method for p∆ (2:1 method)

)()/{0 zLzBBLq ++=∆σ



53

� Case study (p.235)



54



55

� Secondary compression settlement

)/log(loglog 1212 tt

e

tt

eC

∆=

−∆

=α

p

c

p

sct

tH

e

CS log

1)( += α

where pt = time at the end of primary consolidation

')1( αα CeC p+=

αC changes with consolidation stress, so it should be selected based on

present stress level. ( αC for OC soil is quite lower than that for NC soil.)

-



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(%)αC



57

Skempton-Bjerrum modification for consolidation settlement

(Two or three dimensional effect on primary consolidation settlement.)

Conventionally,

1σ∆=∆u

But practically,

)( 313 σσσ ∆−∆+∆=∆ Au

where A=Pore pressure parameter

= f(stress history, soil type, ****)

∫ ∆+∆

=+∆

= )/)1((,

1 0

ue

emdz

e

eS

o

valconvention

∫ ∆= udzmv

dzmv 1σ∆= ∫

where mv= volume coefficient of compressibity

∫ ∆=− udzmS vBS

∫ σ∆−σ∆+σ∆=− dzAmS vBS )]([ 313

Hc 3σ∆

1σ∆

q



58

Settlement Ratio,

∫∫

∆

∆−∆+∆== −

dz

dzA

S

SK

conv

BS

1

313 )]([

σ

σσσ

) ()1(

1

3

ratiosettlementdz

dzAA =

∆

∆−+=

∫∫

σ

σ

For cH =thickness of clay layer,

∫

∫∆

∆−+=

cH

c

dz

dzAAK

H

01

03

)1(

σ

σ

Generally, A < 0.5 for overconsolidated clays.

= 0.5 ~ 1.0 for normally consolidated clays.

> 1.0 for sensitive clays.



59

K is a function of A, shape of foundation and thickness of clay layer,

- Procedure for )( BjerrumSkemptoncS − .

1. Determine )( alconventioncS .

2. Determine A, BH c / .

3. Obtain K from the figure.

4. Calculate )( BjerrumSkemptoncS − .

- Note

K

Documents

10) Foundation SettlementsBased on the theory of elasticity, the elastic settlement of a shallow foundation S e can be defined as, =∫ = ∫∆ − ∆ − ∆ H z s x s y s H Se