Ultimate Bearing Capacitylibvolume3.xyz/.../bearingcapacity/bearingcapacitynotes1.pdf · Foundation Analysis and Design 1 ... occurs is called the ultimate bearing capacity. ... Teng

1

CE-632Foundation Analysis and Design

1

Design

The load per unit area of the foundation at which shear failure in soil occurs 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

men

t qu

2

Settl

e

Sudden or catastrophic failureWell defined failure surfaceBulging on the ground surface adjacent to foundationCommon failure mode in dense sand



Local Shear Failure: Load / Areaq

ttlem

ent

qu

qu1

3

Set

Common in sand or clay with medium compactionSignificant settlement upon loadingFailure surface first develops right below the foundation and then slowly extends outwards with load incrementsFoundation 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.

2



Punching Failure:Load / Area

q

ttlem

ent qu

qu1

4

Set

Common in fairly loose sand or soft clay Failure surface does not extends beyond the zone right beneath the foundationExtensive 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.



Local shear

General shear

unda

tion,

Df/B

*

Relative density of sand, Dr00 0.5 1.0

Vesic (1973)

* 2BLBB L

=+

5

Circular Foundation

Long Rectangular Foundation

Punching shear

Rel

ativ

e de

pth

of fo

u

5

10


Terzaghi’s Bearing Capacity TheoryB

Df

neglected Effective overburdenq = γ’.Df

Strip Footing

a b

j kqu

Rough Foundation Surface

6

Assumption L/B ratio is large plain strain problemDf ≤ BShear resistance of soil for Df depth is neglectedGeneral shear failureShear strength is governed by Mohr-Coulomb Criterion

φ’ φ’45−φ’/2 45−φ’/2

Shear Planes de f

g i

c’- φ’ soilB

I

II IIIII III

3


Terzaghi’s Bearing Capacity Theory

21. 2. 2. .sin tan4u p aq B P C Bφ γ φ′ ′ ′= + −

21. 2. . .sin tan4u pq B P B c Bφ γ φ′ ′ ′ ′= + −

B

qu

7

4p

Iφ’ φ’

ab

dφ’ φ’

Ca= B/2cosφ’

Ca B.tanφ’

Pp Pp

Ppγ = due to only self weight of soil in shear zone

p p pc pqP P P Pγ= + +

Ppc = due to soil cohesion only (soil is weightless)

Ppq = 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 Nγγ ′ . . cB c N . . qB q N

8

2

1 tan 12 cos

PKN γ

γ φφ

⎡ ⎤′= −⎢ ⎥′⎣ ⎦

2

22cos 452

a

qeN

φ=

′⎛ ⎞+⎜ ⎟⎝ ⎠

3 in rad. tan4 2

a π φ φ′⎛ ⎞ ′= −⎜ ⎟

⎝ ⎠( )1 cotc qN N φ′= −

. . 0.5 .u c qq c N q N B Nγγ ′= + +Terzaghi’s bearing capacity equation

Terzaghi’s bearing capacity factors


9

4



Local Shear Failure:

2 . . 0.5 .3u c qq c N q N B Nγγ′ ′ ′ ′ ′= + +

Modify the strength parameters such as: 23mc c′ ′= 1 2tan tan

3mφ φ− ⎛ ⎞′ ′= ⎜ ⎟⎝ ⎠

10

Square and circular footing:

1.3 . . 0.4 .u c qq c N q N B Nγγ′ ′ ′= + +

1.3 . . 0.3 .u c qq c N q N B Nγγ′ ′ ′= + +

For square

For circular


Terzaghi’s Bearing Capacity TheoryEffect of water table:

Dw

Df

Case I: Dw ≤ Df

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

Case II: Df ≤ Dw ≤ (Df + B)

Surcharge, . Fq Dγ=

11

B

B

Limit of influence

In bearing capacity equation replace γ by-

( )w fD DB

γ γ γ γ−⎛ ⎞

′ ′= + −⎜ ⎟⎝ ⎠

Case III: Dw > (Df + B)

No influence of water table.

Another recommendation for Case II:

( ) ( )22 22 w

w sat wdH d H dH H

γγ γ′

= + + −w w fd D D= −

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


Skempton’s Bearing Capacity Analysis for cohesive Soils

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

For strip footing: 5 1 0.2 with limit of 7.5fc c

DN N

B⎛ ⎞

= + ≤⎜ ⎟⎝ ⎠

For square/circular footing:

6 1 0.2 with limit of 9.0fc c

DN N

B⎛ ⎞

= + ≤⎜ ⎟⎝ ⎠

12

g

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

D BN DB L

⎛ ⎞⎛ ⎞= + + ≤⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

7.5 1 0.2 for 2.5c fBN DL

⎛ ⎞= + >⎜ ⎟⎝ ⎠

.u cq c N q= +

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

5


Effective Area Method for Eccentric Loading

B

Df

yx

V

Me

F=

xMe

In case of Moment loading

13

B

eyex

L’=L-2ey

B’=B-2ey

AF=B’L’x

yV

eF

=

In case of Horizontal Force at some 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 γγ= + +

Shape factor

Depth factor

inclination factor

Empirical correction factors

2 φφ ′′⎛ ⎞ ( )1N N φ′ ( ) ( )1 1 4N N φ′

14

2 .tantan 45 .2qN eπ φφ ′⎛ ⎞= +⎜ ⎟

⎝ ⎠( )1 cotc qN N φ′= − ( ) ( )1 tan 1.4qN Nγ 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 γγ= + +

Ground factor Base factor


15

6


Meyerhof’s Correction Factors:

Shape Factors

21 0.2 tan 452c

BsL

φ′⎛ ⎞= + +⎜ ⎟⎝ ⎠

21 0.1 tan 452q

Bs sLγ

φ′⎛ ⎞= = + +⎜ ⎟⎝ ⎠

for 10oφ′ ≥

1qs sγ= =for lower valueφ′

16

Depth Factors 1 0.2 tan 45

2f

c

Dd

Lφ′⎛ ⎞= + +⎜ ⎟

⎝ ⎠ 1 0.1 tan 452

fq

Dd d

Lγφ′⎛ ⎞= = + +⎜ ⎟

⎝ ⎠

for 10oφ′ ≥

1qd dγ= =for lower valueφ′

Inclination Factors

2

190

o

c qi i β⎛ ⎞= = −⎜ ⎟

⎝ ⎠

2

1iγβφ

⎛ ⎞= −⎜ ⎟′⎝ ⎠


Hansen’s Correction Factors:1 for 0

2 .H

cFiBL c

φ′= − =′

( ) 1/211 1 for 0

2 .H

cu

Fi

BL sφ

⎡ ⎤−′= + >⎢ ⎥

⎣ ⎦

For 0

0.4 for fc f

Dd D B

B

φ =

⎡= ≤⎢

⎢

For 0

1 0.4 for fc f

Dd D B

B

φ >

⎡= + ≤⎢

⎢

Inclination Factors

50.51

. .cotH

qV

FiF BL c φ

⎡ ⎤= −⎢ ⎥′ ′+⎣ ⎦

50.71

. .cotH

V

FiF BL cγ φ

⎡ ⎤= −⎢ ⎥′ ′+⎣ ⎦

Depth Factors

Shape Factors

1

0.4 tan for f

c f

Dd D B

B−

⎢⎢

= >⎢⎣11 0.4 tan for f

c f

Dd D B

B−

⎢⎢

= + >⎢⎣

For fD B< For fD B>1dγ =

0.2 . for 0c cBs iL

φ′= =

( )1 . sinq qs i B L φ′= +

( )0.2 1 2 . for 0c cBs iL

φ′= − >

( )1 0.4 .s i B Lγ γ= −

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

( )21 2 tan . 1 sin fq

Dd

Bφ φ

⎛ ⎞′ ′= + − ⎜ ⎟

⎝ ⎠( )2 11 2 tan . 1 sin tan f

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 HB.

3. With a vertical load and a

18

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,Land di,B, di,L. For i,Lsubscripts use ratio L‘/B‘ or D/L‘.

4. Compute qu independently by using (siB, diB) and (siL, diL) and use min value for design.

7


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

19

,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..


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

21

m mL when Hi HL (H parallel to L). If you have both HB and HL use m = (mB

2 + mL2)1/2. Note use

of B and L, not B’, L’.

3. Hi term ≤ 1.0 for computing iq, iγ (always).

8


Suitability of Methods

22


IS:6403-1981 Recommendations

Shape Factors

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 γγ= + − +

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

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

1 0.2cBsL

= + 1 0.2qBsL

= + 1 0.4 BsLγ = −For rectangle,

1 3 1 2

23

1 0.2 tan 452

fc

Dd

Lφ′⎛ ⎞= + +⎜ ⎟

⎝ ⎠

1 0.1 tan 452

fq

Dd d

Lγφ′⎛ ⎞= = + +⎜ ⎟

⎝ ⎠

Inclination Factors

Depth Factors

1.3cs = 1.2qs =0.8 for square, 0.6 for circles sγ 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 21 3 . . 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 and Circular Footing:

For Df > B take Df = B

25

For Df > B, take Df B

[0.5 1 1ww w

f

DR RD

⎛ ⎞= + ≤⎜ ⎟⎜ ⎟

⎝ ⎠

[0.5 1 1w fw 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

nuqqc

D0.5

0

IS:6403-1981 Recommendation:

Cohesionless Soil

26

0 100 200 300 4000

0.0625

B (cm)

1fDB

=1.5B

to 2.0B

Bqc value is taken as

average for this zone

Schmertmann (1975):

2

kg in 0.8 cm

cq

qN Nγ ≅ ≅ ←


Bearing Capacity Correlations with CPT-value

IS:6403-1981 Recommendation:

Cohesive Soil

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

27

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

10


Bearing Capacity of Footing on Layered SoilDepth of rupture zone tan 45

2 2B φ′⎛ ⎞= +⎜ ⎟

⎝ ⎠or approximately taken as “B”

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

B 1

Design using parameters of Layer -1

Case II: Layer-1 is stronger than Layer-2Distribute the stresses to Layer-2 by 2:1 method and check the bearing capacity at this level for

28

B2

1

B

Layer-1

Layer-2

and check the bearing capacity at this level for limit state.

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

Choose minimum value for design

Another approximate method for c‘-φ‘ soil: For effective depth tan 452 2B Bφ′⎛ ⎞+ ≅⎜ ⎟

⎝ ⎠Find average c‘ and φ‘ and use them for ultimate bearing capacity calculation

1 1 2 2 3 3

1 2 3

........av

c H c H c HcH H H

+ + +=

+ + +1 1 2 2 3 3

1 2 3

tan tan tan ....tan....av

H H HH H H

φ φ φφ + + +=

+ + +


Bearing Capacity of Stratified Cohesive SoilIS:6403-1981 Recommendation:

29


Bearing Capacity of Footing on Layered Soil:Stronger Soil Underlying Weaker Soil

Depth “H” is relatively smallPunching shear failure in top layerGeneral shear failure in bottom layer

Depth “H” is relatively largeFull failure surface develops in top layer itself

30

11



31



32

Bearing capacities of continuous footing of with B under vertical load on the surface of homogeneous thick bed of upper and lower soil



For Strip Footing:

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

For Rectangular Footing:

2 11 1

22 tan1 fa su b t

Dc H Kq q H H qB H B

φγ γ′ ′⎛ ⎞

= + + + − ≤⎜ ⎟⎝ ⎠

2 22 tanfDc H KB B φ′ ′⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞

33

2 11 1

22 tan1 1 1 fa su b t

Dc H KB Bq q H H qL B L H B

φγ 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

MQ

max 2

6Q MqBL B L

= +

MeQ

=

min 26Q Mq

BL B L= −

max61Q eq

BL B⎛ ⎞= +⎜ ⎟⎝ ⎠

min61Q eq

BL B⎛ ⎞= −⎜ ⎟⎝ ⎠

34

e

⎝ ⎠

16

eB

>For There will be separation

of foundation from the soil beneath and stresses will be redistributed.

Use for , and B, L for to obtain qu, ,c qd d dγ2B 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 1 and 6 6

L Be eL B

≥ ≥

133

2BeB B

B⎛ ⎞= −⎜ ⎟⎝ ⎠

33⎛ ⎞eB

B1

35

133

2LeL L

L⎛ ⎞= −⎜ ⎟⎝ ⎠

1 112

A L B′ =

ABL

′′ =

′

( )1 1max ,L B L′ =

eL

eB

L1L

B


Case II: 10.5 and 06

L Be eL B

< < <

eL

eB

L1

L2


36

( )1 212

A L L B′ = + ABL

′′ =

′( )1 1max ,L B L′ =

L

B

13



Case III: 1 and 0 0.56

L Be eL B

< < <

eB

B1

37

( )1 212

A L B B′ = + ABL

′′ =

′L L′ =

eL

L

B

B2



Case IV:

B1

e

eB

1 1 and 6 6

L Be eL B

< <

38

ABL

′′ =

′L L′ =

eL

L

B

B2

( )( )2 1 2 212

A L B B B L L′ = + + +



Case V: Circular foundation

eR

39

ALB

′′ =

′

R

R

14


Meyerhof’s (1953) area correction based on empirical correlations: (American Petroleum Institute, 1987)

40


Bearing Capacity of Footings on SlopesMeyerhof’s (1957) Solution

0.5u cq qq c N BNγγ′= +

41

0c′ =Granular Soil

0.5u qq BNγγ=


Bearing Capacity of Footings on SlopesMeyerhof’s (1957) Solution

0φ′ =Cohesive Soil

42

u cqq c N′=

sHNc

γ=

15


Bearing Capacity of Footings on SlopesGraham et al. (1988), Based on method of characteristics

1000

43

For

0fDB

=100

100 10 20 30 40



1000

44

100

100 10 20 30 40

For

0fDB

=



For

0.5fDB

=

45

16



For

1.0fDB

=

46


Bearing Capacity of Footings on SlopesBowles (1997): A simplified approach

B

Df

α α45−φ’/2a c

e

fqu

gα = 45+φ’/2

B

α α

45−φ’/2

a'

b'

c'

e'

g'qu

f'

ror

47

bd

Compute the reduced factor Nc as:

Compute the reduced factor Nq as:

. a b d ec c

abde

LN NL

′ ′ ′ ′′ =

. a e f gq q

aefg

AN N

A′ ′ ′ ′′ =

B

α α45−φ’/2

a'

b'

c'e'

g'qu

d'

f'

b

d'


Soil Compressibility Effects on Bearing CapacityVesic’s (1973) ApproachUse of soil compressibility factors in general bearing capacity equation.These correction factors are function of the rigidity of soil

tans

rvo

GIc σ φ

=′ ′ ′+

Rigidity Index of Soil, Ir:

BCritical Rigidity Index of Soil, Icr:

3.30 0.45

tan 452

BL

φ

⎧ ⎫⎛ ⎞−⎜ ⎟⎪ ⎪⎪ ⎪⎝ ⎠⎨ ⎬′⎡ ⎤⎪ ⎪−⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

48

B/2

( ). / 2vo fD Bσ γ′ = +

20.5.rcI e⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭=

Compressibility Correction Factors, cc, cg, and cq

r rcI I≥For 1c qc c cγ= = =

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

0.6 4.4 .tan1 sin 1

rIBL

qc c eφ

φφ

γ

′⎡ ⎤⎛ ⎞ ′− +⎢ ⎥⎜ ⎟ ′+⎝ ⎠⎣ ⎦= = ≤For

For 0 0.32 0.12 0.60.logc rBc IL

φ′ = → = + +

1For 0

tanq

c qq

cc c

Nφ

φ−

′ > → = −′

Documents

Ultimate Bearing Capacitylibvolume3.xyz/.../bearingcapacity/bearingcapacitynotes1.pdf · Foundation Analysis and Design 1 ... occurs is called the ultimate bearing capacity. ... Teng