6. Settlementof Shallow Footings

CIV4249: Foundation Engineering

Monash University

Oedometer Test

• (change of) Height• Applied Load

• Void Ratio• Applied Stress

Particular Sample Measurements:

General Derived Relationship:

h

height vs time plots

ho

heig

ht

log time

typically take measurements at 15s, 30s,1m, 2m, 3m, 5m, 10m, 15m, 30m, 1h, 2h,3h, 6h, 12h, 24h, 36h, 48h, 60h ….etc.

elastic primaryconsolidation secondary

compression

typically repeat for 12.5, 25, 50, 100, 200, 400, 800 and 1600 KPa

Void ratio = f(h)

RelativeVolume

SpecificGravity

1

e 1.00

2.65

1 + e 1.917

e = 0.8

h = 1.9 cmdia = 6.0 cmW = 103.0 g

Elastic Settlement

• Instantaneous component

• Occurs prior to expulsion of water

• Undrained parameters

• Instantaneous component

• Expulsion of water cannot be separated

• Drained parameters• Not truly elastic

Clay Sand

By definition - fully reversible, no energy loss, instantaneousWater flow is not fully reversible, results in energy loss, and time depends on permeability

Elastic parameters - clay

Eu

• Soft clay• Firm clay• Stiff Clay• V stiff / hard clay

Eu/cu

• most claysnu

• All clays

• 2000 - 5000 kPa• 5000 - 10000 kPa• 10000 - 25000 kPa• 25000 - 60000 kPa

• 200 - 300

• 0.5 (no vol. change)

Elastic parameters - sand

Ed

• Loose sand• Medium sand• Dense sand• V dense sand

nd

• Loose sand• Dense sand

• 10000 - 17000 kPa• 17500 - 25000 kPa• 25000 - 50000 kPa• 50000 - 85000 kPa

• 0.1 to 0.3• 0.3 to 0.4note volume change!

Elastic Settlement

r = H s/E = H.ez

E

s

Hez

Q

Generalized stressand strain field

E

r = ez .dz0

¥

Distribution of Stress

r

R z

Q

sz

sq

sr

• Boussinesq solution

e.g. sz = Q Is z2

Is = 3 1 2p [1+(r/z)2]5/2

Is is stress influence factor

y

Uniformly loaded circular area

dq

dr

r

z

load, q

sz

aBy integration of Boussinesqsolution over complete area:

sz = q [1- 1 ] = q.Is [1+(a/z)2]3/2

Stresses under rectangular area

• Solution after Newmark for stresses under the corner of a uniformly loaded flexible rectangular area:

• Define m = B/z and n = L/z• Solution by charts or

numerically• sz = q.Is

Is = 1 2mn(m2+n2+1)1/2 . m2+n2+2 m2+n2-m2n2+1 4p m2+n2+1

+ tan-12mn(m2+n2+1)1/2

m2+n2-m2n2+1

z

sz

BL

0

1

2

3

4

5

6

7

8

0 0.05 0.1 0.15 0.2 0.25

L/B = 1L/B = 2L/B = 10

Total stress changeIs

z/B

Computation of settlement

1. Determine vertical strains:

r

R z

Q

sz

sq

sr

y

2. Integrate strains:ez = 1 [sz - n ( sr + sq )] Eez = Q .(1+n).cos3y.(3cos2y-2n) 2pz2Er = ez .dz

0

¥

r = Q (1-n2 ) prE

ߥ

â¥

Settlement of a circular area

dq

dr

r

z

load, q

sz

a

Centre :

Edge :

r = 4q(1-n2).apE

r = 2q(1-n2).aE

Settlement at the corner of a flexible rectangular area

z

sz

BL

Schleicher’s solution

r = q.B1 - n2

EIr

Ir = m ln + ln 1p

1+ m2 + 1m

m+ m2 + 1

m = L/B

nz

z

sz= q.I

s

x

Area coveredwith uniformnormal load, q

mzy z

Note: m and n are interchangeable

m = ocm = 3.0m = 2.5

m = 2.0m = 1.8

m = 1.6m = 1.4 m = 1.2

m = 1.0m = 0.9m = 0.8m = 0.7

m = 0.6

m = 0.5

m = 0.4

m = 0.3

m = 0.2

m = 0.1

m = 0.000.01 2 345 0.1 2 43 5 1.0 2 3 45 10

0.26

0.24

0.22

0.20

0.18

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

Is

VERTICAL STRESS BELOW A CORNEROF A UNIFORMLY LOADED FLEXIBLE

RECTANGULAR AREA.

Settlement at the centre of a flexible rectangular area

B

L

B/2

L/2

rcentre = 4q.B 2

1 - n2

EIr Superposition for any

other point under the footing

Settlement under a finite layer - Steinbrenner method

q

H

B

E

“Rigid”

X

Y

rcorner = q.B1 - n2

EIr Ir = F1 + F21-n

1-2n

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

2

4

6

8

10

L/B = 1

L/B = 2

L/B = 5

L/B = 10

L/B = ooL/B = 1

L/B = 2

L/B = 5

L/B = 10L/B = oo

F1

Values of F ( ) and F ( )1 2

Dept

h fa

ctor

d =

H/B

Influence values for settlement beneath the corner of a uniformly loadedrectangle on an elastic layer (Depth D) overlying a rigid base

F2

Superposition using Steinbrenner method

B

L

Multi-layer systemsq

H1B E1

“Rigid”

H2E2

r = r(H1,E1) + r(H1+H2,E2) - r(H1,E2)

• A phenomenon which occurs in both sands and clays

• Can only be isolated as a separate phenomenon in clays

• Expulsion of water from soils accompanied by increase in effective stress and strength

• Amount can be reasonably estimated in lab, but rate is often poorly estimated in lab

• Only partially recoverable

Primary Consolidation

0

1

2

3

4

5

6

7

8

0 0.05 0.1 0.15 0.2 0.25

L/B = 1L/B = 2L/B = 10

Total stress changeIs

z/B

Pore pressure and effective stress changes

s¢i

s¢f

Ds = Du + Ds¢

At t = 0 : Ds = DuAt t = ¥ : Ds = Ds¢

Stress non-linearity

qnet

z

Soil non-linearity

0.40.50.60.70.80.9

11.11.2

10 100 1000

Clay

Cr

Ccp¢cs¢i s¢f

e

sv

r = S log + log Cr H1+eo

Cc H1+ec

p¢c

s¢i

s¢f p¢c

Coeff volume compressibility

0.40.50.60.70.80.9

11.11.2

0 200 400 600 800 1000

Clay

(1+eo).mv

e

sv

r = Smv.Ds¢.DH

Rate of Consolidation

Flowh = H Flowh = H / 2

T = cv ti / H2

U = 90% : T = 0.848

Coefficient of Consolidation

• Coefficient of consolidation, cv (m2/yr)• Notoriously underestimated from

laboratory tests• Determine time required for (90% of)

primary consolidation• Why?

Secondary Compression

• Creep phenomenon• No pore pressure change• Commences at completion of primary

consolidation• ca/Cc » 0.05

ca = Delog (t2 / t1)

r = log (t2/t1)caH

(1+ep)

Flexible vs Rigid

stressstres

sdeflectiondeflection

F F

rcentre 0.8 rcentre RF = 0.8

Depth Correction

0.5

0.6

0.7

0.8

0.9

1

0 2.5 5 7.5 10z/B

Dep

th F

acto

r Bz

Total Settlement

rtot = RF x DF ( relas + rpr.con + rsec )

Field Settlement for Clays(Bjerrum, 1962)

Pore - pressure coefficient

1.2

1.0

0.8

0.6

0.4

0.20 0.2 0.4 0.6 0.8 1.0 1.2

Settl

emen

t co

efficie

nt

Values on curves are DB

0.250.25

4

4

1.0

1.00.5

0.5

Over-consolidated Normallyconsolidated

Verysensitive

clays

CircleStrip

D

B

Clay layer

Differential Settlements

Guiding values• Isolated foundations on clay < 65 mm• Isolated foundations on sand <40 mm

Structural damage to buildings 1/150(Considerable cracking in brick and panel walls)

For the above max settlement valuesflexible structure <1/300rigid structure <1/500

Settlement in Sand via CPT Results (Schmertmann, 1970)

yearsin is 1.0

log2.01

5.01

102

01

121

t

tC

C

zEICCnlayer

layer

z

D¢

DD

ss

sr