6. Settlementof Shallow Footings
6. Settlementof Shallow Footings
CIV4249: Foundation Engineering
Monash University
CIV4249: Foundation Engineering
Monash University
Oedometer TestOedometer Test
• (change of) Height• Applied Load
• Void Ratio• Applied Stress
Particular Sample Measurements:
General Derived Relationship:
h
height vs time plotsheight vs time plots
ho
hei
gh
t
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)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 SettlementElastic 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, instantaneous
Water flow is not fully reversible, results in energy loss, and time depends on permeability
Elastic parameters - clayElastic parameters - clay
Eu
• Soft clay• Firm clay• Stiff Clay• V stiff / hard clay
Eu/cu
• most clays
nu
• All clays
• 2000 - 5000 kPa• 5000 - 10000 kPa• 10000 - 25000 kPa• 25000 - 60000 kPa
• 200 - 300
• 0.5 (no vol. change)
Elastic parameters - sandElastic 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 0.4
note volume change!
Elastic SettlementElastic Settlement
r = H s/E = H.ez
E
s
H
ez
Q
Generalized stressand strain field
E
r = ez .dz0
¥
Distribution of StressDistribution 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 areaUniformly loaded circular area
dq
dr
r
z
load, q
sz
a
By integration of Boussinesqsolution over complete area:
sz = q [1- 1 ] = q.Is [1+(a/z)2]3/2
Stresses under rectangular area
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
B
L
0
1
2
3
4
5
6
7
8
0 0.05 0.1 0.15 0.2 0.25
L/B = 1
L/B = 2
L/B = 10
Total stress changeTotal stress change
Is
z/B
Computation of settlementComputation 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).cos3 .y (3cos2y-2n) 2pz2E
r = ez .dz0
¥
r = Q (1-n2 ) prE
ߥ
â¥
Settlement of a circular areaSettlement of a circular area
dq
dr
r
z
load, q
sz
a
Centre :
Edge :
r = 4q(1-n2).a
pE
r = 2q(1-n2).a
E
Settlement at the corner of a flexible rectangular area
Settlement at the corner of a flexible rectangular area
z
sz
B
L
Schleicher’s solution
r = q.B1 - n2
EIr
Ir = m ln + ln 1p
1+ m2 + 1
mm+ m2 + 1
m = L/B
nz
z
z= q.I
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 = 1.0
m = 0.9
m = 0.8
m = 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
I
VERTICAL STRESS BELOW A CORNEROF A UNIFORMLY LOADED FLEXIBLE
RECTANGULAR AREA.
Settlement at the centre of a 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
Settlement under a finite layer - Steinbrenner method
q
H
B
E
“Rigid”
X
Y
rcorner = q.B1 - n2
EIr Ir = F1 + F2
1-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 = oo
L/B = 1
L/B = 2
L/B = 5
L/B = 10
L/B = oo
F1
Values of F ( ) and F ( )1 2
Dep
th f
acto
r 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 methodSuperposition using Steinbrenner method
B
L
Multi-layer systemsMulti-layer systemsq
H1
BE1
“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 ConsolidationPrimary Consolidation
0
1
2
3
4
5
6
7
8
0 0.05 0.1 0.15 0.2 0.25
L/B = 1
L/B = 2
L/B = 10
Total stress changeTotal stress change
Is
z/B
Pore pressure and effective stress changes
Pore pressure and effective stress changes
¢s i
¢s f
Ds = Du + Ds¢
At t = 0 : Ds = DuAt t = ¥ : Ds = Ds¢
Stress non-linearityStress non-linearity
qnet
z
Soil non-linearitySoil non-linearity
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
10 100 1000
Clay
Cr
Ccp¢c¢s i ¢s f
e
sv
r = S log + log Cr H
1+eo
Cc H
1+ec
p¢c
¢s i
¢s f
p¢c
Coeff volume compressibilityCoeff volume compressibility
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
0 200 400 600 800 1000
Clay
(1+eo).mv
e
sv
r = Smv.Ds¢.DH
Rate of ConsolidationRate of Consolidation
Flowh = H Flowh = H / 2
T = cv ti / H2
U = 90% : T = 0.848
Coefficient of ConsolidationCoefficient of Consolidation
• Coefficient of consolidation, cv (m2/yr)
• Notoriously underestimated from laboratory tests
• Determine time required for (90% of) primary consolidation
• Why?
Secondary CompressionSecondary Compression
• Creep phenomenon• No pore pressure change• Commences at completion of primary
consolidation
• ca/Cc » 0.05
ca = De
log (t2 / t1)r = log (t2/t1)
caH
(1+ep)
Flexible vs RigidFlexible vs Rigid
stressstres
sdeflectiondeflection
F F
rcentre 0.8 rcentre RF = 0.8RF = 0.8
Depth CorrectionDepth 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 SettlementTotal Settlement
rtot = RF x DF ( relas + rpr.con + rsec )
Field Settlement for Clays(Bjerrum, 1962)
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
Settle
men
t co
effici
ent
Values on curves areDB
0.25
0.25
4
4
1.0
1.00.5
0.5
Over-consolidatedNormally
consolidated
Verysensitive
clays
CircleStrip
D
B
Clay layer
Differential SettlementsDifferential 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 values
flexible structure <1/300
rigid structure <1/500
Settlement in Sand via CPT Results (Schmertmann, 1970)Settlement in Sand via CPT
Results (Schmertmann, 1970)
yearsin is
1.0log2.01
5.01
102
01
121
t
tC
C
zE
ICC
nlayer
layer
z