Ultimate Bearing Capacity

Transcription

CE-632
Foundation Analysis and
Design
Ultimate Bearing Capacity
The load per unit area of the foundation at which shear failure in soil
occurs is called the ultimate bearing capacity.
1
Foundation Analysis and Design: Dr. Amit Prashant
Principal Modes of Failure:
General Shear Failure:
Load / Area
q
Settlement
qu

Sudden or catastrophic failure
Well defined failure surface
Bulging on the ground surface adjacent to foundation
Common failure mode in dense sand
2
Load / Area
q
Local Shear Failure:
Setttlement
qu1

qu
Common in sand or clay with medium compaction
Significant settlement upon loading
Failure surface first develops right below the foundation and then
slowly extends outwards with load increments
Foundation movement shows sudden jerks first (at qu1) and then
after a considerable amount of movement the slip surface may
reach the ground.
A small amount of bulging may occur next to the foundation.
3
1
Load / Area
q
Punching Failure:
qu1
Setttlement
qu
Common in fairly loose sand or soft clay
Failure surface does not extends beyond the zone right beneath the
foundation
Extensive settlement with a wedge shaped soil zone in elastic
equilibrium beneath the foundation. Vertical shear occurs around the
edges of foundation.
After reaching failure load-settlement curve continues at some slope
and mostly linearly.

4
Relative depth of fou
undation, Df/B*
0
Vesic (1973)
Relative density of sand, Dr
0.5
0
1.0
General
shear
Local
shear
B* =
2BL
B+L
Circular
Foundation
5
Punching
shear
Long
Rectangular
Foundation
10
5
Terzaghi’s Bearing Capacity Theory
B
Rough Foundation
Surface
Strip Footing
k
j
Effective overburden
q = γ’.Df
qu
neglected
Df
a
g
45−φ’/2
b
φ’
I
φ’
III
Shear
Planes
II
II
e
d
45−φ’/2
i
III
c’- φ’ soil
f
Assumption
L/B ratio is large Æ plain strain problem
Df ≤ B
Shear resistance of soil for Df depth is neglected
General shear failure
Shear strength is governed by Mohr-Coulomb Criterion
B
6
2
B
1
qu .B = 2.Pp + 2.Ca .sin φ ′ − γ ′B 2 tan φ ′
4
qu
b
a
φ’
Ca= B/2
cosφ’
φ’
Pp
1
qu .B = 2.Pp + B.c′.sin φ ′ − γ ′B 2 tan φ ′
4
φ’
I
Pp = Ppγ + Ppc + Ppq
Ca B.tanφ’
Ppγ = due to only self weight of soil
in shear zone
φ’
d
Pp
Ppc = due to soil cohesion only
(soil is weightless)
Ppq = due to surcharge only
7
Weight term
Cohesion term
1
⎛
⎞
qu .B = ⎜ 2.Ppγ − γ ′B 2 tan φ ′ ⎟ + ( 2.Ppc + B.c′.sin φ ′ ) + 2.Ppq
4
⎝
⎠
B. ( 0.5γ ′B.Nγ )
Surcharge term
B.c.Nc
B.q.N q
Terzaghi’s bearing
capacity equation
qu = c.N c + q.N q + 0.5γ ′B.Nγ
Terzaghi’s bearing capacity factors
Nγ =
⎡ K
⎤
1
tan φ ′ ⎢ P2γ − 1⎥
′
φ
2
cos
⎣
⎦
N c = ( N q − 1) cot φ ′
e2 a
φ′ ⎞
⎛
2 cos 2 ⎜ 45 + ⎟
2⎠
⎝
⎛ 3π φ ′ in rad. ⎞
−
a=⎜
⎟ tan φ ′
2
⎝ 4
⎠
Nq =
8
9
3
Local Shear Failure:
2
cm′ = c′
3
Modify the strength parameters such as:
⎛2
⎝
⎞
⎠
φm′ = tan −1 ⎜ tan φ ′ ⎟
3
2
qu = c′.N c′ + q.N q′ + 0.5γ ′B.Nγ′
3
Square and circular footing:
qu = 1.3c′.N c + q.N q + 0.4γ ′B.Nγ′
For square
qu = 1.3c′.N c + q.N q + 0.3γ ′B.Nγ′
For circular
10
Effect of water table:
Case I: Dw ≤ Df
Surcharge, q = γ .Dw + γ ′ ( D f − Dw )
Dw
Case II: Df ≤ Dw ≤ (Df + B)
Df
Surcharge, q = γ .DF
In bearing capacity equation
replace γ by-
B
⎛ Dw − D f ⎞
⎟ (γ − γ ′)
B
⎝
⎠
Case III: Dw > (Df + B)
γ =γ′+⎜
B
Limit of influence
No influence of water table.
Another recommendation for Case II:
γ = ( 2H + dw )
dw
γ′
2
γ sat + 2 ( H − d w )
H2
H
d w = Dw − D f
Rupture depth: H = 0.5 B tan ( 45 + φ ′ 2 )
11
Skempton’s Bearing Capacity Analysis for
cohesive Soils
~ For saturated cohesive soil, φ‘ = 0 Æ
N q = 1, and Nγ = 0
Df ⎞
⎛
For strip footing:
N c = 5 ⎜1 + 0.2
⎟ with limit of N c ≤ 7.5
B ⎠
⎝
D ⎞
⎛
N c = 6 ⎜1 + 0.2 f ⎟ with limit of N c ≤ 9.0
B ⎠
⎝
For square/circular
g
footing:
For rectangular footing:
D ⎞⎛
⎛
B⎞
N c = 5 ⎜1 + 0.2 f ⎟⎜1 + 0.2 ⎟ for D f ≤ 2.5
B ⎠⎝
L⎠
⎝
B⎞
⎛
N c = 7.5 ⎜1 + 0.2 ⎟ for D f > 2.5
L⎠
⎝
qu = c.N c + q
Net ultimate bearing capacity,
qnu = qu − γ .D f
qu = c.N c
12
4
Effective Area Method for Eccentric Loading
In case of Moment loading
Df
B
AF=B’L’
B’=B-2ey
L’=L-2ey
ex
ey
ex =
My
ey =
Mx
FV
FV
In case of Horizontal Force at
some height but the column is
centered on the foundation
M y = FHx .d FH
M x = FHy .d FH
13
General Bearing Capacity Equation:
(Meyerhof, 1963)
qu = c.N c .sc .d c .ic + q.N q .sq .d q .iq + 0.5γ .B.Nγ .sγ .dγ .iγ
Shape
factor
Depth
factor
φ′ ⎞
⎛
N q = tan 2 ⎜ 45 + ⎟ .eπ .tan φ ′
2⎠
⎝
inclination
factor
Empirical correction
factors
N c = ( N q − 1) cot φ ′
Nγ = ( N q − 1) tan (11.4
4φ ′ )
[By Hansen(1970):
N γ = 1.5 ( N q − 1) tan (φ ′ )
[By Vesic(1973):
Nγ = 2 ( N q + 1) tan (φ ′ )
qu = c.N c .sc .dc .ic .gc .bc + q.N q .sq .d q .iq .g q .bq + 0.5γ .B.Nγ .sγ .dγ .iγ .gγ .bγ
Ground factor
Base factor
14
15
5
Meyerhof’s Correction Factors:
Shape
Factors
sc = 1 + 0.2
B
φ′ ⎞
⎛
tan 2 ⎜ 45 + ⎟
L
2⎠
⎝
for φ ′ ≥ 10o
sq = sγ = 1 + 0.1
B
φ′ ⎞
⎛
tan 2 ⎜ 45 + ⎟
L
2⎠
⎝
for lower φ ′ value
sq = sγ = 1
Depth
Factors
d c = 1 + 0.2
φ′ ⎞
⎛
tan ⎜ 45 + ⎟
L
2⎠
⎝
Df
for φ ′ ≥ 10o
d q = dγ = 1 + 0.1
Df
L
φ′ ⎞
⎛
tan ⎜ 45 + ⎟
2⎠
⎝
for lower φ ′ value
d q = dγ = 1
Inclination
Factors
⎛ βo ⎞
ic = iq = ⎜ 1 −
⎟
⎝ 90 ⎠
2
⎛ β⎞
iγ = ⎜1 − ⎟
⎝ φ′ ⎠
2
16
Hansen’s Correction Factors:
Inclination
Factors
Depth
Factors
FH
for φ ′ = 0
2 BL.c′
5
⎡
⎤
0.5 FH
iq = ⎢1 −
⎥
′
′
F
BL
.
c
.cot
φ
+
V
⎣
⎦
ic = 1 −
1/2
For φ = 0
For φ > 0
Df
⎡
for D f ≤ B
⎢ d c = 0.4
B
⎢
D
⎢
f
−1
⎢⎣ d c = 0.4 tan B for D f > B
Df
⎡
for D f ≤ B
⎢ d c = 1 + 0.4
B
⎢
D
⎢
f
−1
⎢⎣ d c = 1 + 0.4 tan B for D f > B
For D f < B
2 ⎛ Df ⎞
d q = 1 + 2 tan φ ′. (1 − sin φ ′ ) ⎜
⎟
⎝ B ⎠
Shape
Factors
1 ⎡ (1 − FH ) ⎤
⎢1 +
⎥ for φ ′ > 0
2⎣
BL.su ⎦
5
⎡
⎤
0.7 FH
iγ = ⎢1 −
⎥
′
′
⎣ FV + BL.c .cot φ ⎦
ic =
sc = 0.2ic .
B
L
for φ ′ = 0
sq = 1 + iq . ( B L ) sinφ ′
For D f > B
⎛ Df ⎞
2
d q = 1 + 2 tan φ ′. (1 − sin φ ′ ) tan −1 ⎜
⎟
⎝ B ⎠
dγ = 1
B
for φ ′ > 0
L
sγ = 1 − 0.4iγ . ( B L )
sc = 0.2 (1 − 2ic ) .
Hansen’s Recommendation for cohesive saturated soil, φ'=0 Æ
qu = c.Nc . (1 + sc + dc + ic ) + q
Notes:
1. Notice use of “effective”
base dimensions B‘, L‘ by
Hansen but not by Vesic.
2. The values are consistent
with a vertical load or a
vertical load accompanied by
a horizontal load HB.
3. With a vertical load and a
load HL (and either HB=0 or
HB>0) you may have to
compute two sets of shape
and depth factors si,B, si,L
and di,B, di,L. For i,L
subscripts use ratio L‘/B‘ or
D/L‘.
4. Compute qu independently
by using (siB, diB) and (siL,
diL) and use min value for
design.
18
6
Notes:
1. Use Hi as either HB or HL,
or both if HL>0.
2. Hansen (1970) did not give
an ic for φ>0. The value given
here is from Hansen (1961)
and also used by Vesic.
3. Variable ca = base
adhesion,, on the order of 0.6
to 1.0 x base cohesion.
4. Refer to sketch on next
slide for identification of
angles η and β , footing depth
D, location of Hi (parallel and
at top of base slab; usually
also produces eccentricity).
Especially notice V = force
normal to base and is not the
resultant R from combining V
and Hi..
19
20
Note:
1. When φ=0 (and β≠0) use
Nγ = -2sin(±β) in Nγ term.
2. Compute m = mB when
Hi = HB (H parallel to B) and
m = mL when Hi = HL (H
parallel to L). If you have
both HB and HL use
m = (mB2 + mL2)1/2. Note use
of B and L, not B’, L’.
3. Hi term ≤ 1.0 for
computing iq, iγ (always).
21
7
Suitability of Methods
22
IS:6403-1981 Recommendations
Net Ultimate Bearing capacity:
qnu = c.Nc .sc .dc .ic + q. ( N q − 1) .sq .dq .iq + 0.5γ .B.Nγ .sγ .dγ .iγ
qnu = cu .N c .sc .d c .ic
For cohesive soils Æ
N c , N q , Nγ
Shape
Factors
sc = 1 + 0.2
For rectangle,
For square and circle,
Depth
Factors
Inclination
Factors
where,
B
L
sq = 1 + 0.2
B
L
sγ = 1 − 0.4
B
L
12
sc = 1.3
1 3 sq = 1.2
sγ = 0.8 for square, sγ = 0.6 for circle
φ′ ⎞
⎛
tan ⎜ 45 + ⎟
2⎠
⎝
Df
φ′ ⎞
⎛
tan ⎜ 45 + ⎟
d q = dγ = 1 + 0.1
2⎠
L
⎝
d q = dγ = 1 for φ ′ < 10o
d c = 1 + 0.2
N c = 5.14
as per Vesic(1973) recommendations
Df
L
The same as Meyerhof (1963)
for
φ ′ ≥ 10o
23
Bearing Capacity
Correlations with
SPT-value
Peck, Hansen, and
Thornburn (1974)
&
IS:6403-1981
Recommendation
24
8
Bearing Capacity Correlations with SPT-value
Teng (1962):
(
)
(
)
1⎡
3 N ′′2 .B.Rw′ + 5 100 + N ′′2 .D f .Rw ⎦⎤
6⎣
For Strip Footing:
qnu =
For Square and
Circular Footing:
1
qnu = ⎡⎣ N ′′2 .B.Rw′ + 3 100 + N ′′2 .D f .Rw ⎤⎦
3
For Df > B,
B take Df = B
Dw
Water Table Corrections:
⎛ D ⎞
Rw = 0.5 ⎜1 + w ⎟
⎜ Df ⎟
⎝
⎠
⎛ Dw − D f ⎞
Rw′ = 0.5 ⎜1 +
⎟
⎜
D f ⎟⎠
⎝
[ Rw ≤ 1
Df
B
[ Rw′ ≤ 1
B
Limit of influence
25
Bearing Capacity Correlations with CPT-value
0. 2500
IS:6403-1981 Recommendation:
Cohesionless Soil
qnu
qc
0.1675
0
0.1250
0.5
Df
B
1.5B
to
2.0B
0.0625
qc value is
taken as
average for
this zone
B
0
0
100
200
300
400
B (cm)
Schmertmann (1975):
Nγ ≅ N q ≅
=1
qc
0.8
← in
kg
cm 2
26
Bearing Capacity Correlations with CPT-value
Cohesive Soil
qnu = cu .N c .sc .dc .ic
Soil Type
Point Resistance Values
( qc ) kgf/cm2
Range of Undrained
Cohesion (kgf/cm2)
Normally consolidated
clays
qc < 20
qc/18 to qc/15
Over consolidated clays
qc > 20
qc/26 to qc/22
27
9
Bearing Capacity of Footing on Layered Soil
Depth of rupture zone =
B
φ′ ⎞
⎛
tan ⎜ 45 + ⎟ or approximately taken as “B”
2
2⎠
⎝
Case I: Layer-1 is weaker than Layer-2
Design using parameters of Layer -1
Case II: Layer-1 is stronger than Layer-2
Layer-1
B
Distribute the stresses to Layer-2 by 2:1 method
and check the bearing capacity at this level for
limit state.
1
B
2
Also check the bearing capacity for original
foundation level using parameters of Layer-1
Layer-2
Choose minimum value for design
Another approximate method for c‘-φ‘ soil: For effective depth
B
φ′ ⎞
⎛
tan ⎜ 45 + ⎟ ≅ B
2
2⎠
⎝
Find average c‘ and φ‘ and use them for ultimate bearing capacity calculation
cav =
c1 H1 + c2 H 2 + c3 H 3 + ....
H1 + H 2 + H 3 + ....
tan φav =
tan φ1 H1 + tan φ2 H 2 + tan φ3 H 3 + ....
H1 + H 2 + H 3 + ....
28
Bearing Capacity of Stratified Cohesive Soil
29
Bearing Capacity of Footing on Layered Soil:
Stronger Soil Underlying Weaker Soil

Depth “H” is relatively small
Punching shear failure in top layer
General shear failure in bottom
layer

Depth “H” is relatively large
Full failure surface develops in top
layer itself
30
10
Bearing Capacity of
Footing on Layered Soil:
Stronger Soil Underlying
Weaker Soil
31
Bearing capacities of continuous footing of with B
under vertical load on the surface of homogeneous
thick bed of upper and lower soil
32
For Strip Footing:
qu = qb +
⎛ 2 D f ⎞ K s tan φ1′
2ca′ H
+ γ 1H 2 ⎜1 +
− γ 1 H ≤ qt
⎟
B
H ⎠
B
⎝
Where, qt is the bearing capacity for foundation considering
only the top layer to infinite depth
For Rectangular Footing:
⎛ B ⎞ ⎛ 2c′ H
qu = qb + ⎜1 + ⎟ ⎜ a
⎝ L ⎠⎝ B
B ⎞ ⎛ 2 D f ⎞ K s tan φ1′
⎞
2⎛
− γ 1 H ≤ qt
⎟ + γ 1 H ⎜1 + L ⎟ ⎜1 + H ⎟
B
⎝
⎠⎝
⎠
⎠
Special Cases:
1. Top layer is strong sand and bottom layer is saturated soft clay
c′1 = 0 φ2 = 0
2. Top layer is strong sand and bottom layer is weaker sand
c′1 = 0
c′2 = 0
2. Top layer is strong saturated clay and bottom layer is weaker saturated clay
φ1 = 0
φ2 = 0
33
11
Eccentrically Loaded Foundations
Q
M
e=
M
Q
qmax =
Q 6M
+
BL B 2 L
qmax =
Q ⎛ 6e ⎞
⎜1 + ⎟
BL ⎝
B⎠
qmin =
Q 6M
−
BL B 2 L
qmin =
Q ⎛ 6e ⎞
⎜1 − ⎟
BL ⎝
B⎠
B
For
e
e 1
There will be separation
>
B 6
of foundation from the soil beneath
and stresses will be redistributed.
B′ = B − 2e
for
L′ = L
Use
sc , sq , sγ , and B, L for d c , d q , dγ
to obtain qu
The effective area method for two way eccentricity becomes
a little more complex than what is suggested above.
It is discussed in the subsequent slides
Qu = qu . A′
34
Determination of Effective Dimensions for Eccentrically
Loaded foundations (Highter and Anders, 1985)
Case I:
eL 1
e
1
≥
and B ≥
L 6
B 6
⎛ 3 3e ⎞
B1 = B ⎜ − B ⎟
⎝2 B ⎠
B1
eB
L
eL
L1
⎛ 3 3e ⎞
L1 = L ⎜ − L ⎟
⎝2 L ⎠
A′ =
B
1
L1 B1
2
B′ =
L′ = max ( B1 , L1 )
A′
L′
35
Determination of Effective Dimensions for Eccentrically Loaded
foundations (Highter and Anders, 1985)
Case II:
L2
eL
e
1
< 0.5 and 0 < B <
L
B 6
eB
eL
L1
L
B
1
( L1 + L2 ) B
2
L′ = max ( B1 , L1 )
A′ =
B′ =
A′
L′
36
12
Case III: eL < 1 and 0 < eB < 0.5
6
L
B
B1
eB
eL
L
B
B2
1
A′ = L ( B1 + B2 )
A′
2
B′ =
L′
L′ = L
37
Case IV:
eL 1
e
1
and B <
<
L 6
B 6
B1
eB
eL
L
B
B2
1
A′ = L2 B + ( B1 + B2 )( L + L2 )
2
A′
L′ = L
B′ =
L′
38
Case V: Circular foundation
eR
R
L′ =
A′
B′
39
13
Meyerhof’s (1953) area correction based on empirical
correlations: (American Petroleum Institute, 1987)
40
Bearing Capacity of
Footings on Slopes
Meyerhof’s (1957)
Solution
qu = c′N cq + 0.5γ BN γ q
Granular Soil
c′ = 0
qu = 0.5γ BN γ q
41
Bearing Capacity of
Footings on Slopes
Meyerhof’s (1957)
Solution
Cohesive Soil
φ′ = 0
qu = c′N cq
Ns =
γH
c
42
14
Bearing Capacity of
Footings on Slopes
Graham et al. (1988),
Based on method of
characteristics
1000
For
Df
100
10
B
0
20
10
30
=0
40
43
Bearing Capacity of
Footings on Slopes
Graham et al. (1988),
Based on method of
characteristics
1000
For
Df
100
10
B
0
10
20
40
30
=0
44
Bearing Capacity of Footings on Slopes
Graham et al. (1988), Based on method of characteristics
For
Df
B
= 0.5
45
15
Graham et al. (1988), Based on method of characteristics
For
Df
= 1.0
B
46
Bowles (1997): A simplified approach
B
f
B
α = 45+φ’/2
g'
f'
g
qu
qu
Df
a
45−φ’/2
e
α
a'
c
α
α
e'
45−φ’/2
c'
ro
r
b'
b
b
d
α
d'
B

g'
qu
N c′ = N c .
f'
a'
e'
Compute the reduced factor Nc as:
α
c'
α

45−φ’/2
Compute the reduced factor Nq as:
N q′ = N q .
b'
d'
La′b′d ′e′
Labde
Aa′e′f ′g ′
Aaefg
47
Soil Compressibility Effects on Bearing Capacity
Vesic’s (1973) Approach

Use of soil compressibility factors in general bearing capacity equation.
These correction factors are function of the rigidity of soil
Rigidity Index of Soil, Ir:
Ir =
Gs
′ tan φ ′
c′ + σ vo
Critical Rigidity Index of Soil, Icr:
I rc = 0.5.e
⎧
B ⎞⎫
⎛
3.30 − ⎜ 0.45 ⎟ ⎪
L ⎠⎪
⎪⎪
⎝
⎨
⎬
⎪ tan ⎡ 45 − φ ′ ⎤ ⎪
⎢
2 ⎦⎥ ⎭⎪
⎣
⎩⎪
B
B/2
σ vo′ = γ . ( D f + B / 2 )
Compressibility Correction Factors, cc, cg, and cq
For
I r ≥ I rc
cc = cq = cγ = 1
For
I r < I rc
cq = cγ = e ⎣⎝
⎡⎛
3.07.sin φ ′.log10 ( 2. I r ) ⎤
B
⎞
⎢⎜ 0.6 − 4.4 ⎟.tan φ ′ +
⎥
L
1+ sin φ ′
⎠
⎦
For φ ′ = 0 → cc = 0.32 + 0.12
For φ ′ > 0 → cc = cq −
1 − cq
≤1
B
+ 0.60.log I r
L
N q tan φ ′
48
16

Ultimate Bearing Capacity

Transcription

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