CE 632 Bearing Capacity Handout

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1

CE-632 Foundation Analysis and

1

The load per unit area of the foundation at which shear failure in soiloccurs is called the ultimate bearing capacity.

Ultimate Bearing Capacity

Foundation Analysis and Design: Dr. Amit Prashant

Principal Modes of Failure:

General Shear Failure: Load / Areaq

m e n t qu

2

S e t t l e

Sudden or catastrophic failure

Well defined failure surface

Bulging on the ground surface adjacent to foundation

Common failure mode in dense sand



Local Shear Failure:Load / Area

q

t l e m e n t

qu

qu1

3

S e

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 mayreach the ground.

A small amount of bulging may occur next to the foundation.



2



Punching Failure:Load / Area

q

t l e m e n t qu

qu1

4

S e

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 theedges of foundation.

After reaching failure load-settlement curve continues at some slopeand mostly linearly.



Localshear

Generalshear

u n d a t i o n ,

D f / B *

Relative density of sand, Dr

00

0.5 1.0

Vesic (1973)

* 2BLB

B L=

+

5

CircularFoundation

LongRectangularFoundation

Punchingshear

R e l a t i v e d e p

t h o f f o

5

10


Terzaghi’s Bearing Capacity Theory B

Df

neglected Effective overburdenq =γ’.Df

Strip Footing

a b

j k

qu

Rough FoundationSurface

6

Ass ump tio n

L/B ratio is largeÆ 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

φ’ φ’ 45 −φ’/2 45 −φ’/2

ShearPlanes d e f

g i

c’ - φ’ soil B

I

II II

II I II I



3


Terzaghi’s Bearing Capacity Theory

21. 2. 2. .sin tan

4

u p aq B P C Bφ γ φ ′ ′ ′= + −

21. 2. . .sin tanu pq B P B c Bφ γ φ ′ ′ ′ ′= + −

B

qu

7

I’ φ’

ab

d’ φ’

Ca=B/2cosφ’

Ca B.tanφ’

P p P p

P pγ =due to only self weight of soilin shear zone

p p pc pqP P P Pγ = + +

P pc =due to soil cohesion only(soil is weightless)

P pq =due to surcharge only



( )21. 2. tan 2. . .sin 2.

4u p pc pqq B P B P B c Pγ γ φ φ

⎛ ⎞′ ′ ′ ′= − + + +⎜ ⎟⎝ ⎠

Weight term Cohesion term

Surcharge term

( ). 0.5 .B B Nγ γ ′ . . cBc N . . qBq N

8

2

1tan 1

2 cos

PK N

γ

γ φ φ

⎡ ⎤′= −⎢ ⎥′⎣ ⎦

2

22cos 452

a

q

eN

φ =

′⎛ ⎞+⎜ ⎟⎝ ⎠

3 in rad.tan

4 2a

π φ φ

′⎛ ⎞ ′= −⎜ ⎟⎝ ⎠

( )1 cotc qN N φ ′= −

. . 0.5 .u c qq c N q N B Nγ γ ′= + + Terzaghi’s bearingcapacity equation

Terzaghi’s bearing capacity factors


9



4



Local Shear Failure:

2. . 0.5 .

3u c qq c N qN B Nγ γ ′ ′ ′ ′ ′= + +

Modify the strength parameters such as: 2

3mc c′ ′=

1 2

tan tan3mφ φ

− ⎛ ⎞

′ ′= ⎜ ⎟⎝ ⎠

10

Square and circular footing:

1.3 . . 0.4 .u c qq c N qN B Nγ γ ′ ′ ′= + +

1.3 . . 0.3 .u c qq c N qN B Nγ γ ′ ′ ′= + +

For square

For circular



Effect of water table:

Dw

Df

Case I: Dw ≤ D f

Surcharge, ( ). w f wq D D Dγ γ ′= + −

Case II: D f ≤ Dw ≤ (Df + B)

Surcharge, . Fq Dγ =

11

B

B

Limit of influence

In bearing capacity equationreplace γ by-

( )w f

D D

Bγ γ γ γ

−⎛ ⎞′ ′= + −⎜ ⎟⎝ ⎠

Case III: Dw > (Df + B)

No influence of water table.

Another recommendation for Case II:

( ) ( )2

2 22 w

w sat w

dH d H d

H H

γ γ γ

′= + + −

w w f d D D= −

( )0.5 tan 45 2H B φ ′= +Rupture depth:


Skempton’s Bearing Capacity Analysis f or cohesive Soils

~For saturated cohesive soil, φ‘ =0 Æ 1, and 0qN Nγ = =

For strip footing: 5 1 0.2 with limit of 7.5f

c c

DN N

B

⎛ ⎞= + ≤⎜ ⎟

⎝ ⎠

For square/circularfooting:

6 1 0.2 with limit of 9.0f

c c

DN N

B

⎛ ⎞= + ≤⎜ ⎟

⎝ ⎠

12

For rectangular footing: 5 1 0.2 1 0.2 for 2.5f

c f

D BN D

B L

⎛ ⎞⎛ ⎞= + + ≤⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

7.5 1 0.2 for 2.5c f

BN D

L

⎛ ⎞= + >⎜ ⎟⎝ ⎠

.u cq c N q= +

Net ultimate bearing capacity, .nu u f q q Dγ = − .u cq c N=



5


Effective Area Method for Eccentric Loading

Df

yx

V

MeF

=

xM

In case of Moment loading

13

ey e x

L’=L-2ey

B’=B-2ey

AF =B’L’ y

VF=

In case of Horizontal Force atsome height but the column is

centered on the foundation

.y Hx FHM F d=

.x Hy FHM F d=


General Bearing Capacity Equation:

(Meyerhof, 1963)

. . . . . . . . 0.5 . . . . .u c c c c q q q qq c N s d i q N s d i B N s d iγ γ γ γ γ = + +

Shapefactor

Depthfactor

inclinationfactor

Empirical correctionfactors

′ ′ ′

14

2 .tantan 45 .2

qN eπ ′= +⎜ ⎟⎝ ⎠

cotc q φ = − tan .qγ φ = −

( ) ( )2 1 tanqN Nγ φ ′= +

( ) ( )1.5 1 tanqN Nγ φ ′= −

[By Vesic(1973):

[By Hansen(1970):

. . . . . . . . . . . . 0.5 . . . . . . .u c c c c c c q q q q q qq c N s d i g b q N s d i g b B N s d i g bγ γ γ γ γ γ = + +

Ground factor Base factor


15



6


Meyerhof’s Correction Factors:

ShapeFactors

21 0.2 tan 452

c

Bs

L

φ ′⎛ ⎞= + +⎜ ⎟⎝ ⎠

21 0.1 tan 452

q

Bs s

Lγ

φ ′⎛ ⎞= = + +⎜ ⎟⎝ ⎠

for 10oφ ′ ≥

1qs sγ = =for lower valueφ ′

16

DepthFactors 1 0.2 tan 45

2

f

c

Dd

L

φ ′⎛ ⎞= + +⎜ ⎟⎝ ⎠ 1 0.1 tan 45

2

f

q

Dd d

Lγ

φ ′⎛ ⎞= = + +⎜ ⎟⎝ ⎠

for 10oφ ′ ≥

1qd dγ = =

for lower valueφ ′

InclinationFactors

2

190

o

c qi iβ ⎛ ⎞

= = −⎜ ⎟⎝ ⎠

2

1iγ

β

φ

⎛ ⎞= −⎜ ⎟′⎝ ⎠


Hansen’s Correction Factors:

1 for 02 .

Hc

Fi

BLcφ ′= − =

′( )

1/211

1 for 02 .

H

c

u

Fi

BL sφ

⎡ ⎤−′= + >⎢ ⎥

⎣ ⎦

For 0

0.4 forf

c f

Dd D B

B

φ =

⎡= ≤⎢

For 0

1 0.4 forf

c f

Dd D B

B

φ >

⎡= + ≤⎢

InclinationFactors

5

0.51

. .cotH

q

V

Fi

F BL c φ

⎡ ⎤= −⎢ ⎥′ ′+⎣ ⎦

5

0.71

. .cotH

V

Fi

F BL cγ

φ

⎡ ⎤= −⎢ ⎥′ ′+⎣ ⎦

DepthFactors

ShapeFactors

1

0.4tan forf

c f

Dd D B

B

−⎢= >⎢⎣

11 0.4tan forf

c f

Dd D B

B

−⎢= + >⎢⎣

For f D B< For f D B> 1dγ =

0.2 . for 0c c

Bs i

Lφ ′= =

( )1 . sinq qs i B L φ ′= +

( )0.2 1 2 . for 0c c

Bs i

Lφ ′= − >

( )1 0.4 .s i B Lγ γ = −

Hansen’s Recommendation for cohesive saturated soil, φ '=0 Æ ( ). . 1u c c c cq c N s d i q= + + + +

( )2

1 2tan . 1 sinf

q

Dd

Bφ φ

⎛ ⎞′ ′= + − ⎜ ⎟

⎝ ⎠( )

2 11 2tan . 1 sin tanf

q

Dd

Bφ φ

− ⎛ ⎞′ ′= + − ⎜ ⎟⎝ ⎠


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 H B.

3. With a verticalload anda

18

.

load H L (and either H B=0 or H B>0) you may have tocompute two sets of shapeand depth factors si,B, si,L

and d i,B, d i,L. For i ,Lsubscripts use ratio L‘/B‘ or D/L‘.

4. Compute qu independently by using (siB, d iB ) and (siL,d iL ) and use min value for design.



7


Notes:

1. Use H i as either H B or H L,or both if H L>0.

2. Hansen (1970) did not givean i c for φ >0. The value given

here is from Hansen (1961)and also used by Vesic.

3. Variable c a = baseadhesion, on the order of 0.6

19

to 1.0 x base cohesion.

4. Refer to sketch on next slide for identification of

angles η and β , footing depthD, location of H i (parallel and at top of base slab; usually also produces eccentricity).

Especially notice V = forcenormal to base and is not theresultant R from combining V and H i ..


20


Note:

1. When φ =0 (and β ≠0) useN γ = -2sin(± β ) in N γ term.

2. Compute m = mB whenH i = H B (H parallel to B) and m = m when H = H H

21

L i L

parallel to L). If you haveboth H B and H L usem = (mB

2 + mL2 )1/2 . Note use

of B and L, not B’, L’.

3. H i term ≤ 1.0 for computing i q, i γ (always).



8


Suitability of Methods

22


IS:6403-1981 Recommendat ions

ShapeFactors

Net Ultimate Bearing capacity: ( ). . . . . 1 . . . 0.5 . . . . .nu c c c c q q q qq c N s d i q N s d i B N s d iγ γ γ γ γ = + − +

. . . .nu u c c c cq c N s d i= 5.14cN =For cohesive soilsÆ where,

, ,c qN N Nγ as per Vesic(1973) recommendations

1 0.2c

Bs

L= + 1 0.2q

Bs

L= + 1 0.4

Bs

Lγ

= −For rectangle,

23

1 0.2 tan 452

f

cDdL

φ ′⎛ ⎞= + +⎜ ⎟⎝ ⎠

1 0.1 tan 452

f

q

Dd d

Lγ

φ ′⎛ ⎞= = + +⎜ ⎟⎝ ⎠

InclinationFactors

DepthFactors

.cs = .qs =0.8 for square, 0.6 for circles s

γ γ = =For square and circle,

for 10oφ ′ ≥

1qd dγ

= = for 10oφ ′ <

The same as Meyerhof (1963)


Bearing Capacity Correlations with

SPT-value

Peck, Hansen, and

Thornburn (1974)

&

IS:6403-1981

24

Recommendation



9


Bearing Capacity Correlations with SPT-value

Teng (1962):

( )2 213 . . 5 100 . .

6nu w f wq N B R N D R⎡ ⎤′′ ′ ′′= + +⎣ ⎦

For Strip Footing:

( )2 21. . 3 100 . .

3nu w f wq N B R N D R⎡ ⎤′′ ′ ′′= + +⎣ ⎦

For Square andCircular Footing:

=

25

,

[0.5 1 1ww w

f

DR R

D

⎛ ⎞= + ≤⎜ ⎟⎜ ⎟

⎝ ⎠

[0.5 1 1w f

w w

f

D DR R

D

⎛ ⎞−′ ′= + ≤⎜ ⎟⎜ ⎟

⎝ ⎠

Water Table Corrections:

B

B

Dw

Df

Limit of influence


Bearing Capacity Correlations with CPT-value

0.1250

0.1675

0. 2500

nuq

qc0.5

0

IS:6403-1981 Recomm endation :

Cohesionless Soil

26

0 100 200 300 4000

0.0625

B (cm)

1f

B=

1.5B

to2.0B

Bqc value is

taken asaverage forthis zone

Schmertm ann (1975):

2

kgin

0.8 cmc

q

qN N

γ ≅ ≅ ←


Bearing Capacity Correlations with CPT-value

IS:6403-1981 Recomm endation :

Cohesive Soil

. . . .nu u c c c cq c N s d i=

27

Soil TypePoint Resistance Values

( qc ) kgf/cm2

Range of UndrainedCohesion (kgf/cm2)

Normally consolidatedclays

qc <20 qc/18 to qc/15

Over consolidated clays qc >20 qc/26 to qc/22



10


Bearing Capacity o f Footing on Layered Soil

Depth of rupture zone tan 452 2

B φ ′⎛ ⎞= +⎜ ⎟⎝ ⎠

or approximately taken as “B”

Case I: Layer-1 is weaker than Layer-2

1

Design using parameters of Layer -1

Case II: Layer-1 is stronger than Layer-2

Distribute the stresses to Layer-2 by 2:1 method

28

2

B

Layer-1

Layer-2

limit state.

Also check the bearing capacity for originalfoundation level using parameters of Layer-1

Choose minimum value for design

Ano ther approxi mate metho d fo r c‘ -φ‘ soil: For effective depth tan 452 2

BB

φ ′⎛ ⎞+ ≅⎜ ⎟⎝ ⎠

Find average c‘ andφ ‘ and use them for ultimate bearing capacity calculation

1 1 2 2 3 3

1 2 3

....

....av

c H c H c Hc

H H H

+ + +=

+ + +1 1 2 2 3 3

1 2 3

tan tan tan ....tan

....av

H H H

H H H

φ φ φ φ

+ + +=

+ + +


Bearing Capacity of Stratified Cohesive Soil IS:6403-1981 Recommendatio n:

29


Bearing Capacity of Foot ing on Layered Soil:Stronger Soil Underlying Weaker Soil

Depth “H” is relatively small

Punching shear failure in top layer

General shear failure in bottomlayer

Depth “H” is relatively large

Full failure surface develops in toplayer itself

30



11


Bearing Capacity of

Footing on Layered Soil:

Stronger Soil Underlying

Weaker Soil

31


Bearing Capacity of Footing on Layered Soil:

Stronger Soil Underlying Weaker Soil

32

Bearing capacities of continuous footing of with B

under vertical load on the surface of homogeneousthick bed of upper and lower soil


Bearing Capacity of Foot ing on Layered Soil:Stronger Soil Underlying Weaker Soil

For Strip Footing:

Where, qt is the bearing capacity for foundation consideringonly the top layer to infinite depth

For Rectangular Footing:

2 11 1

22 tan1

f a su b t

Dc H K q q H H q

B H B

φ γ γ

′ ′⎛ ⎞= + + + − ≤⎜ ⎟

⎝ ⎠

2D′ ′

33

2 11 11 1 1a s

u b tq q H H qL B L H B

γ γ = + + + + + − ≤⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

Special Cases:

1. Top layer is strong sand and bottom layer is saturated soft clay

2 0φ =

2. Top layer is strong sand and bottom layer is weaker sand

1 0c′ =

1 0c′ = 2 0c′ =

2. Top layer is strong saturated clay and bottom layer is weaker saturated clay

2 0φ =1 0φ =



12


Eccentrically Loaded Foundations

B

M

Q

max 2

6Q M

q BL B L= +

Me

Q=

min 2

6Q Mq

BL B L= −

max

6

1

Q e

q BL B

⎛ ⎞= +⎜ ⎟⎝ ⎠

min

61

Q eq

BL B

⎛ ⎞= −⎜ ⎟⎝ ⎠

34

e1

6

e

B>For There will be separation

of foundation from the soil beneathand stresses will be redistributed.

Use for , and B, L for to obtain qu, ,c qd d dγ

2B B e′ = −

L L′ =, ,c qs s sγ

.u uQ q A′= 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


Determination of Effective Dimensions for Eccentrically

Loaded foundations (Highter and Anders, 1985)

Case I: 1 1and

6 6L Be e

L B≥ ≥

1

33

2Be

B BB

⎛ ⎞= −⎜ ⎟⎝ ⎠

e

B1

35

12

LeL L

L= −⎜ ⎟

⎝ ⎠

1 112

A L B′ =

AB

L

′′ =

′

( )1 1max ,L B L′ =

eL L1L

B


Case II:1

0.5 and 06

L Be e

L B< < <

eL

eB

L1

L2

Determination of Effective Dimensions f or Eccentrically Loaded

foundations (Highter and Anders, 1985)

36

( )1 2

1

2A L L B′ = + A

BL

′′ =

′( )1 1max ,L B L′ =

L

B



13




Case III: 1and 0 0.5

6L Be e

L B< < <

eB

B1

37

( )1 2

1

2A L B B′ = +

AB

L

′′ =

′L L′ =

eL

L

B

B2




Case IV:

B1

eB

1 1and

6 6L Be e

L B< <

38

AB

L

′′ =

′L L′ =

L

L

B

B2

( ) ( )2 1 2 2

1

2A L B B B L L′ = + + +




Case V: Circular foundation

eR

39

AL

B

′′ =

′

R



14


Meyerhof’s (1953) area correcti on based on emp irical

correlations: (American Petroleum Institute, 1987)

40


Bearing Capacity of

Footings on Slopes

Meyerhof’s (1957)

Solution

0.5u cq qq c N BNγ γ ′= +

41

0c′ =Granular Soil

0.5u qq BNγ γ =


Bearing Capacity of

Footings on Slopes

Meyerhof’s (1957)

Solution

0φ ′ =Cohesive Soil

42

u cqq c N′=

s

H

N c

γ

=



15


Bearing Capacity of

Footings on Slopes

Graham et al. (1988),

Based on method of

characteristics

1000

43

For

0f D

B=

100

100 10 20 30 40


Bearing Capacity of

Footings on Slopes

Graham et al. (1988),

Based on method of

characteristics

1000

44

100

100 10 20 30 40

For

0f D

B=


Bearing Capacity of Foot ings on Slopes

Graham et al. (1988), Based on method of characteristics

For

0.5f D

B=

45





Graham et al. (1988), Based on method of characteristics

For

1.0f D

B=

46



Bowles (1997): A simpl ified approach

B

Df

α α45 −φ’/2

a c

e

f qu

gα = 45+φ’/2

B

α α

45 −φ’/2

a'

'

c'

e'

g'qu

f'

ror

47

bd

Compute the reduced factor Nc as:

Compute the reduced factor Nq as:

. a bdec c

abde

LN N

L′ ′ ′ ′′ =

.a e fg

q q

aefg

AN N

A

′ ′ ′ ′′ =

B

α α

45 −φ’/2

a'

b'

c'

e'

g'qu

d'

f'

d'


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

tans

r

vo

GI

c σ φ =

′ ′ ′+Rigidity Index of Soil, I r :

BCritical Rigidity Index of Soil, I cr :

3.30 0.45

tan 45

B

L

φ

⎧ ⎫⎛ ⎞−⎜ ⎟⎪ ⎪⎪ ⎪⎝ ⎠⎨ ⎬′⎡ ⎤⎪ ⎪−⎢ ⎥

48

B/2

( ). /2vo f D Bσ γ ′ = +

0.5.rcI e⎢ ⎥

=

Compressibility Correction Factors, c c , c g , and c q

r rcI I≥For 1c qc c cγ

= = =

r rcI I<( )103.07.sin .log 2.

0.6 4.4 .tan1 sin

1

rIB

L

qc c e

φ φ

φ

γ

′⎡ ⎤⎛ ⎞ ′− +⎢ ⎥⎜ ⎟ ′+⎝ ⎠⎣ ⎦= = ≤For

For 0 0.32 0.12 0.60.logc r

Bc I

Lφ ′ = → = + +

1For 0 tan

qc q

q

cc c Nφ

φ

−′ > → = − ′

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CE 632 Bearing Capacity Handout

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