Post on 05-Jan-2016
Bearing Capacity
ظرفيت باربري
Footing
Shallow Foundations
Footing
Shallow Foundations
Shallow Foundations
D
B
QTypical Buried Footing
Shallow Foundations
D
B
Q
qs = DQ
Typical Buried Footing
Equivalent Surface Footing
Shallow Foundations
D
B
Q
qs = DQ
Typical Buried Footing
Equivalent Surface Footing
Shallow Foundations have D/B < 1
Methods of analysis
•Lower bound approach–failure stress state in equilibrium
–failure load less than or equal to true collapse
•Upper bound approach–failure mechanism assumed
–failure load greater than or equal to true collapse
Shallow Foundations
Shallow Foundations
q f
Footing
Surcharge q s
Shallow Foundations
q f
Footing
Surcharge q s
H
Frictionless Discontinuity
Shallow Foundations
q f
Footing
Surcharge q s
Soil at stateof ActiveFailure with v > h
H
Frictionless Discontinuity
Shallow Foundations
q f
Footing
Surcharge q s
Soil at stateof ActiveFailure with v > h
H
Frictionless Discontinuity 1 3 2 N c N
Shallow Foundations
q f
Footing
Surcharge q s
Soil at stateof ActiveFailure with v > h
Soil at stateof PassiveFailure with h > v
H
Frictionless Discontinuity 1 3 2 N c N
Shallow Foundations
q f
Footing
Surcharge q s
Soil at stateof ActiveFailure with v > h
Soil at stateof PassiveFailure with h > v
H
Frictionless Discontinuity 1 3 2 N c N N
c
c
1
3
cot
cot
Shallow Foundations
v = 1
h = 3
h = 1
v = 3
Shallow Foundations
v fq z
v = 1
h = 3
h = 1
v = 3
v sq z
Shallow Foundations
v fq z
v = 1
h = 3
Nq z c
cf
h
cot
cot
h = 1
v = 3
v sq z
Nc
q z ch
s
cot
cot
Shallow Foundations
v fq z
v = 1
h = 3
Nq z c
cf
h
cot
cot
h fNq z c c
1( cot ) cot
h = 1
v = 3
v sq z
Nc
q z ch
s
cot
cot
h sN q z c c ( cot ) cot
Shallow Foundations
( ) ( ) h active h passive
HH
dz dz 00
Shallow Foundations
( ) ( ) h active h passive
HH
dz dz 00
1
2 2
2 2
Nq H
Hc H N q H
Hc Hf s
cot cot
Shallow Foundations
( ) ( ) h active h passive
HH
dz dz 00
1
2 2
2 2
Nq H
Hc H N q H
Hc Hf s
cot cot
q q NH
N c Nf s
2 2 2
21 1cot
•This solution will give a lower bound to the true solution because of the simplified stress distribution assumed in the soil
•Similar terms occur in all bearing capacity expressions. They are functions of the friction angle and
•the surcharge applied to the soil surface
•the self weight of the soil
•cohesion
Shallow Foundations
q q NH
N c Nf s
2 2 2
21 1cot
•A general bearing capacity equation can be written
Shallow Foundations
q q NB
N c Nf s q c
2
•A general bearing capacity equation can be written
•The terms Nq, N and Nc are known as the bearing capacity factors
Shallow Foundations
q q NB
N c Nf s q c
2
•A general bearing capacity equation can be written
•The terms Nq, N and Nc are known as the bearing capacity factors
•Values can be determined from charts
Shallow Foundations
q q NB
N c Nf s q c
2
Shallow Foundations
BEARING CAPACITY FACTORS [After Terzaghi and Peck (1948)]
60 50 40 30 20 10 0 20 40 60 80
N and N
0
10
20
30
40
(deg
rees
)
q c N
NN q
B
Da
bc
d
q= D
Q f
ff
Bearing capacity of a shallow foundation
ULTIMATE BEARING CAPACITY OF CLAY ( = 0 only) (After A.W. Skempton)
0 1 2 3 4 5
D/B
5
6
7
8
9
Nc
5.14
B
D
N (for rectangle)
= (0.84+0.16 ) N (square)
L= Length of footing
BL c
q = cNcult
q = B N + cN + D N continuous footing12f c f q
q = 0.4 BN + 1.3cN + D N squaref c f q
q = 0.6 RN + 1.3cN + D N circularf c f q
q = cN + Df c
c
Nc
BEARING CAPACITY THEORIES OF TERZAGHI AND SKEMPTON
Dq= D
Qf
ff
B
Shallow Foundations
Mechanism analysed by Terzaghi
Effect of Foundation Shape
q q NB
N c Nf s q c
2
Continuous strip footing
Effect of Foundation Shape
q q NB
N c Nf s q c
2
q q N BN c Nf s q c 0 13.4 .
Continuous strip footing
Square footing
Effect of Foundation Shape
q q NB
N c Nf s q c
2
q q N BN c Nf s q c 0 13.4 .
q q N BN c Nf s q c 0 6 13. .
Continuous strip footing
Square footing
Circular footing
•Effective stress analysis is needed to assess the long term foundation capacity .
•Total and effective stresses are identical if the soil is dry. The analysis is identical to that described above except that the parameters used in the equations are c´, ´, dry
rather than cu, u, sat .
•If the water table is more than a depth of 1.5 B (the footing width) below the base of the footing the water can be assumed to have no effect.
Effective Stress Analysis
•If the soil below the base of the footing is saturated, the analysis must account for the water pressures.
Effective Stress Analysis
•If the soil below the base of the footing is saturated, the analysis must account for the water pressures.
Effective Stress Analysis
qs = D
Q = q f B
u = u o
•If the soil below the base of the footing is saturated, the analysis must account for the water pressures.
Effective Stress Analysis
qs = D
Q = q f B
u = u o
•If the soil below the base of the footing is saturated, the analysis must account for the water pressures.
Effective Stress Analysis
qs = D
Q = q f B
u = u o
•If the soil below the base of the footing is saturated, the analysis must account for the water pressures.
Effective Stress Analysis
qs = D
Q = q f B
u = u o
Effective Stress Analysis
These effective quantities are required because Mohr Coulomb failure criterion must be expressed in terms of effective stress
Nc
c
1
3
cot
cot
Effective Stress Analysis
These effective quantities are required because Mohr Coulomb failure criterion must be expressed in terms of effective stress
Nc
c
1
3
cot
cot
The total vertical stress, pore pressure and effective vertical stress at any depth z beneath the footing are
Effective Stress Analysis
These effective quantities are required because Mohr Coulomb failure criterion must be expressed in terms of effective stress
Nc
c
1
3
cot
cot
The total vertical stress, pore pressure and effective vertical stress at any depth z beneath the footing are
Effective Stress Analysis
These effective quantities are required because Mohr Coulomb failure criterion must be expressed in terms of effective stress
Nc
c
1
3
cot
cot
The total vertical stress, pore pressure and effective vertical stress at any depth z beneath the footing are
Effective Stress Analysis
v fq z
’v = ’1
’h = ’3
Nq z c
cf
h
cot
cot
h fN
q z c c1
( cot ) cot
v sq z
Nc
q z ch
s
cot
cot
h sN q z c c( cot ) cot
’h = ’1
’v = ’3
Effective Stress Analysis
q q NH
N c Nf s
2 2 2
21 1cot
The simple analysis leads to
Effective Stress Analysis
q q NH
N c Nf s
2 2 2
21 1cot
The simple analysis leads to
This is similar to the previous expression except that now all terms involve effective quantities.
Effective Stress Analysis
q q NH
N c Nf s
2 2 2
21 1cot
q q NB
N c Nf s q c
2
The simple analysis leads to
This is similar to the previous expression except that now all terms involve effective quantities.
As before a general expression can be written with the form
Effective Stress Analysis
q q NH
N c Nf s
2 2 2
21 1cot
q q NB
N c Nf s q c
2
The simple analysis leads to
This is similar to the previous expression except that now all terms involve effective quantities.
As before a general expression can be written with the form
The Bearing Capacity Factors are identical to those from Total Stress Analysis
Effective Stress Analysis
q q NH
N c Nf s
2 2 2
21 1cot
q q NB
N c Nf s q c
2
The simple analysis leads to
This is similar to the previous expression except that now all terms involve effective quantities.
As before a general expression can be written with the form
The Bearing Capacity Factors are identical to those from Total Stress Analysis
Note that the Total Bearing Capacity qf = q’f + uo
Analysis has so far considered
•soil strength parameters
•rate of loading (drained or undrained)
•groundwater conditions (dry or saturated)
•foundation shape (strip footing, square or circle)
Other important factors include
•soil compressibility
•embedment (D/B > 1)
•inclined loading
•eccentric loading
•non-homogeneous soil
Effective Stress Analysis
•More theoretically accurate bearing capacity factors are given on pages 69 to 71 of the Data Sheets
•In practice the Terzaghi factors are still widely used .
•The bearing capacity equation assumes that the effects of c', , and ' can be superimposed .
•This is not correct as there is an interaction between the three effects because of the plastic nature of the soil response.
Effective Stress Analysis
•The formulae give the ultimate bearing capacity
•Significant deformations and large settlements may occur before general bearing failure occurs
•Local failure (yield) will occur at some depth beneath the footing at a load less than the ultimate collapse load
•The zone of plastic (yielding) soil will then spread as the load is increased. Only when the failure zone extends to the surface will a failure mechanism exist.
•A minimum load factor of 3 against ultimate failure is usually adopted to keep settlements within acceptable bounds, and to avoid problems with local failure.
Effective Stress Analysis
Example
D = 2m
B = 5mQ
Example
D = 2m
B = 5mQ
Determine short term and long term ultimate capacity given
Example
qs
Q=q f B
Equivalent surface footing
Example
qs
Q=q f B
Equivalent surface footing
Short term - Undrained (total stress) analysis
Example
qs
Q=q f B
Equivalent surface footing
Short term - Undrained (total stress) analysis
Position of water table not important - soil must be saturated
Example
qs
Q=q f B
Equivalent surface footing
Short term - Undrained (total stress) analysis
Position of water table not important - soil must be saturated
Example
BEARING CAPACITY FACTORS [After Terzaghi and Peck (1948)]
60 50 40 30 20 10 0 20 40 60 80
N and N
0
10
20
30
40
(deg
rees
)
q c N
NN q
B
Da
bc
d
q= D
Q f
ff
Bearing capacity of a shallow foundation
ULTIMATE BEARING CAPACITY OF CLAY ( = 0 only) (After A.W. Skempton)
0 1 2 3 4 5
D/B
5
6
7
8
9
Nc
5.14
B
D
N (for rectangle)
= (0.84+0.16 ) N (square)
L= Length of footing
BL c
q = cNcult
q = B N + cN + D N continuous footing12f c f q
q = 0.4 BN + 1.3cN + D N squaref c f q
q = 0.6 RN + 1.3cN + D N circularf c f q
q = cN + Df c
c
Nc
BEARING CAPACITY THEORIES OF TERZAGHI AND SKEMPTON
u = 0 Nq = 1, N = 0 and Nc = 5.14
Example
Short term capacity
q q NB
N c Nf s q c
2
Example
Short term capacity
q q NB
N c Nf s q c
2
q f = 30 1 + 0 + 25 5.14 = 158.5 kPa (Bearing capacity)
Example
Short term capacity
q q NB
N c Nf s q c
2
q f = 30 1 + 0 + 25 5.14 = 158.5 kPa (Bearing capacity)
Q = q f B = 158.5 5 = 792.5 kN/m (Bearing Force)
Example
Long term capacity
Effective stress (fully drained) analysis
Example
Long term capacity
Effective stress (fully drained) analysis
Example
Long term capacity
Effective stress (fully drained) analysis
Example
Long term capacity
Effective stress (fully drained) analysis
Example
Long term capacity
Effective stress (fully drained) analysis
Example
BEARING CAPACITY FACTORS [After Terzaghi and Peck (1948)]
60 50 40 30 20 10 0 20 40 60 80
N and N
0
10
20
30
40
(deg
rees
)
q c N
NN q
B
Da
bc
d
q= D
Q f
ff
Bearing capacity of a shallow foundation
ULTIMATE BEARING CAPACITY OF CLAY ( = 0 only) (After A.W. Skempton)
0 1 2 3 4 5
D/B
5
6
7
8
9
Nc
5.14
B
D
N (for rectangle)
= (0.84+0.16 ) N (square)
L= Length of footing
BL c
q = cNcult
q = B N + cN + D N continuous footing12f c f q
q = 0.4 BN + 1.3cN + D N squaref c f q
q = 0.6 RN + 1.3cN + D N circularf c f q
q = cN + Df c
c
Nc
BEARING CAPACITY THEORIES OF TERZAGHI AND SKEMPTON
’ = 25 Nq = 13, N = 10 and Nc = 24.5
Example
Long term capacity
Example
Long term capacity
Example
Long term capacity
Total Stress Analysis u = 0
f c u sq = N c + q
Total Stress Analysis u = 0
f c u sq = N c + q
BEARING CAPACITY FACTORS [After Terzaghi and Peck (1948)]
60 50 40 30 20 10 0 20 40 60 80
N and N
0
10
20
30
40
(deg
rees
)
q c N
NN q
B
Da
bc
d
q= D
Q f
ff
Bearing capacity of a shallow foundation
ULTIMATE BEARING CAPACITY OF CLAY ( = 0 only) (After A.W. Skempton)
0 1 2 3 4 5
D/B
5
6
7
8
9
Nc
5.14
B
D
N (for rectangle)
= (0.84+0.16 ) N (square)
L= Length of footing
BL c
q = cNcult
q = B N + cN + D N continuous footing12f c f q
q = 0.4 BN + 1.3cN + D N squaref c f q
q = 0.6 RN + 1.3cN + D N circularf c f q
q = cN + Df c
c
Nc
BEARING CAPACITY THEORIES OF TERZAGHI AND SKEMPTON
Bottom heave into excavations
D
B
heave
Bottom heave into excavations
D
Bottom heave into excavations
D
Bottom heave into excavations
D
Bottom heave into excavations
D