6. Settlement
of Shallow Footings
CIV4249: Foundation Engineering
Monash University
Oedometer Test
Particular Sample
Measurements:
General Derived
Relationship:
• (change of) Height
• Applied Load
• Void Ratio
• Applied Stress
h
height
height vs time plots
ho
typically take measurements at 15s, 30s,
1m, 2m, 3m, 5m, 10m, 15m, 30m, 1h, 2h,
3h, 6h, 12h, 24h, 36h, 48h, 60h ….etc.
elastic
primary
consolidation
secondary
compression
typically repeat for 12.5, 25, 50, 100, 200, 400, 800 and 1600 KPa
log time
Void ratio = f(h)
e
1.00
e = 0.8
1
2.65
Relative
Volume
Specific
Gravity
1+e
1.917
h = 1.9 cm
dia = 6.0 cm
W = 103.0 g
Elastic Settlement
By definition fully reversible, no energy loss, instantaneous
Water flow is not fully reversible, results in
energy loss, and time depends on permeability
Clay
• Instantaneous
component
• Occurs prior to
expulsion of water
• Undrained parameters
Sand
• Instantaneous
component
• Expulsion of water
cannot be separated
• Drained parameters
• Not truly elastic
Elastic parameters - clay
•
•
•
•
•
Eu
Soft clay
Firm clay
Stiff Clay
V stiff / hard clay
Eu/cu
most clays
•
•
•
•
2000 - 5000 kPa
5000 - 10000 kPa
10000 - 25000 kPa
25000 - 60000 kPa
• 200 - 300
nu
• All clays
• 0.5 (no vol. change)
Elastic parameters - sand
•
•
•
•
Ed
Loose sand
Medium sand
Dense sand
V dense sand
•
•
•
•
10000 - 17000 kPa
17500 - 25000 kPa
25000 - 50000 kPa
50000 - 85000 kPa
nd
• Loose sand
• Dense sand
• 0.1 to 0.3
• 0.3 to 0.4
note volume change!
Elastic Settlement
Q
s
H
E
E
ez
r = H s/E = H.ez
Generalized stress
and strain field

r=
0
ez .dz
Distribution of Stress
Q
• Boussinesq solution
e.g. sz = Q Is
z2
R
sz
Is is stress influence factor
Is = 3
1
2p [1+(r/z)2]5/2
y
r
sr
sq
z
Uniformly loaded circular area
load, q
dr
By integration of
Boussinesq
solution over
complete area:
sz = q [1-
dq
a
1
] = q.Is
[1+(a/z)2]3/2
r
z
sz
Stresses under rectangular
area
L
• 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
4p
B
sz
z
2+n2+1)1/2
2mn(m
2mn(m2+n2+1)1/2 . m2+n2+2
-1
+
tan
m2+n2-m2n2+1
m2+n2+1
m2+n2-m2n2+1
Total stress change
0
0 .0 5
0 .1
0 .1 5
0 .2
0
1
2
3
z/B
4
5
L /B = 1
L /B = 2
6
L /B = 1 0
7
8
0 .2 5
Is
Computation of settlement
Q
1. Determine vertical strains:


2. Integrate strains:
ez = 1 [sz - n ( sr + sq )]
E
ez = Q .(1+n).cos3y.(3cos2y-2n)
2pz2E

r=
0
r=
ez .dz
Q (1-n2 )
prE
R
y
z
sz
r
sr
sq


Settlement of a circular area
load, q
dr
Centre :
r=
2q(1-n2).a
dq
a
r
E
Edge :
z
r = 4q(1-n2).a
pE
sz
Settlement at the corner of a
flexible rectangular areaL
Schleicher’s solution
r = q.B
1 - n2
E
Ir
sz
z
m = L/B
Ir = 1 m ln
p
B
1+ m2 + 1
m
+ ln m+ m2 + 1
0.26
Is
A rea covered
w ith uniform
norm al load,
q
0.24
0.22
nz
m = 3 .0
m = 2 .5
m = 2 .0
m = 1 .8
m = 1 .6
m = 1 .2
m = 1 .4
m = 1 .0
x
0.20
mz
y
z
m = 0 .9
sz =
q.I
0.18
m = oc
m = 0 .8
s
m = 0 .7
z
0.16
0.14
m = 0 .6
N ote: m and n are interchangeable
m = 0 .5
0.12
m = 0 .4
0.10
m = 0 .3
0.08
m = 0 .2
0.06
0.04
m = 0 .1
0.02
0
0.01
m = 0 .0
2
3 4 5
0.1
2
3 4 5
1.0
2
3 45
V E R T IC A L S T R E S S B E LO W A C O R N E R
O F A U N IF O R M LY LO AD E D F LE X IB LE
R E C TA N G U LA R A R E A .
10
Settlement at the centre of a
flexible rectangular area
L
L/2
B
B/2
rcentre = 4q.B
2
1 - n2
E
Ir
Superposition for any other
point under the footing
Settlement under a finite layer Steinbrenner method
rcorner = q.B
1 - n2
E
Ir
q
X
B
H
E
Y
“Rigid”
Ir = F1 +
1-2n
1-n
F2
Va lu es o f F 1 (
0
0 .1
0 .2
0.3
0 .4
) and F 2 (
0 .5
)
0 .6
0 .7
0 .8
L /B = 1
D epth factor d = H /B
2
L /B = 2
F1
4
L /B = 5
F2
6
L /B = 5
L /B = 1 0
8
L /B = oo
L /B = 2
L /B = 1
L /B = 1 0
L /B = oo
10
Influe nce va lue s for se ttle m e nt be ne a th the corne r of a uniform ly loa de d
re cta ngle on a n e la stic la ye r (D e pth D ) ove rlying a rigid ba se
Superposition using
Steinbrenner method
L
B
Multi-layer systems
q
H1
r = r(H1,E1) + r(H1+H2,E2) - r(H1,E2)
B
E1
E2
H2
“Rigid”
Primary Consolidation
• 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
Total stress change
0
0 .0 5
0 .1
0 .1 5
0 .2
0
1
2
3
z/B
4
5
L /B = 1
L /B = 2
6
L /B = 1 0
7
8
0 .2 5
Is
Pore pressure and effective
stress changes
Ds = Du + Ds
At t = 0 : Ds = Du
At t =  : Ds = Ds
sf
si
Stress non-linearity
qnet
z
pc
sf
Cr H
Cc H
r = S 1+eo log s + 1+ec log p
i
c
Soil non-linearity
Cr
1 .2
1 .1
1
0 .9
e
0 .8
0 .7
C la y
0 .6
si
0 .5
0 .4
10
pc
100
sf
Cc
1000
sv
Coeff volume compressibility
r = Smv.Ds.DH
1 .2
1 .1
1
C la y
0 .9
e
0 .8
0 .7
0 .6
(1+eo).mv
0 .5
0 .4
0
200
400
600
800
1000
sv
Rate of Consolidation
h h= =HH/ 2
T = cv ti / H2
U = 90% : T = 0.848
Flow
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 =
De
log (t2 / t1)
r=
caH
(1+ep)
log (t2/t1)
Flexible vs Rigid
F
stress
deflection
rcentre
F
stres
s
deflection
0.8 rcentre
RF = 0.8
Depth Correction
1
D e p th F a c to r
0 .9
z
0 .8
0 .7
0 .6
0 .5
0
2 .5
5
7 .5
10
z/B
B
Total Settlement
rtot = RF x DF ( relas + rpr.con + rsec )
Field Settlement for Clays
(Bjerrum, 1962)
1.2
S ettlem en t coefficien t 
Values on curves are
D
B
1.0
0.25
0.25
0.8
B
0.5
4
1.0
0.6
C lay
layer
0.5
D
4
1.0
0.4
C ircle
Po re - p ressu re co
efficien t
S trip
N orm ally
consolidated
O ver-consolidated
Very
sensitive
clays
0.2
0
0.2
0.4
0.6
0.8
1.0
1.2
Differential Settlements
Guiding values
• Isolated foundations on clay
• Isolated foundations on sand
< 65 mm
<40 mm
Structural damage to buildings
1/150
(Considerable cracking in brick and panel walls)
For the above max settlement values
flexible structure
<1/300
rigid structure
<1/500
Settlement in Sand via CPT
Results (Schmertmann, 1970)
layer  n
r  C 1C 2 D s

layer  1
Iz
Dz
E
 s 0 
C 1  1  0 .5

 Ds 
 t 
C 2  1  0 . 2 log 10 

 0 .1 
t is in years