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

Foundation Analysis and Design: Dr. Amit Prashant

Principal Modes of Failure: 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

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Foundation Analysis and Design: Dr. Amit Prashant

Principal Modes of Failure: 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.

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Foundation Analysis and Design: Dr. Amit Prashant

Relative depth of fou undation, Df/B*

Principal Modes of Failure: 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

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Foundation Analysis and Design: Dr. Amit Prashant

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

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Foundation Analysis and Design: Dr. Amit Prashant

Terzaghi’s Bearing Capacity Theory 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

Foundation Analysis and Design: Dr. Amit Prashant

Terzaghi’s Bearing Capacity Theory 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 =

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Foundation Analysis and Design: Dr. Amit Prashant

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Foundation Analysis and Design: Dr. Amit Prashant

Terzaghi’s Bearing Capacity Theory 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

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Foundation Analysis and Design: Dr. Amit Prashant

Terzaghi’s Bearing Capacity Theory 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 )

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Foundation Analysis and Design: Dr. Amit Prashant

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

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Foundation Analysis and Design: Dr. Amit Prashant

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

Foundation Analysis and Design: Dr. Amit Prashant

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

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Foundation Analysis and Design: Dr. Amit Prashant

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Foundation Analysis and Design: Dr. Amit Prashant

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

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Foundation Analysis and Design: Dr. Amit Prashant

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

Foundation Analysis and Design: Dr. Amit Prashant

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

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Foundation Analysis and Design: Dr. Amit Prashant

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

Foundation Analysis and Design: Dr. Amit Prashant

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Foundation Analysis and Design: Dr. Amit Prashant

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).

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Foundation Analysis and Design: Dr. Amit Prashant

Suitability of Methods

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Foundation Analysis and Design: Dr. Amit Prashant

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

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Foundation Analysis and Design: Dr. Amit Prashant

Bearing Capacity Correlations with SPT-value Peck, Hansen, and Thornburn (1974) & IS:6403-1981 Recommendation

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Foundation Analysis and Design: Dr. Amit Prashant

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

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Foundation Analysis and Design: Dr. Amit Prashant

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

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Foundation Analysis and Design: Dr. Amit Prashant

Bearing Capacity Correlations with CPT-value IS:6403-1981 Recommendation: 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

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Foundation Analysis and Design: Dr. Amit Prashant

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 + ....

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Foundation Analysis and Design: Dr. Amit Prashant

Bearing Capacity of Stratified Cohesive Soil IS:6403-1981 Recommendation:

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Foundation Analysis and Design: Dr. Amit Prashant

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

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Foundation Analysis and Design: Dr. Amit Prashant

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

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Foundation Analysis and Design: Dr. Amit Prashant

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

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

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Foundation Analysis and Design: Dr. Amit Prashant

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

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Foundation Analysis and Design: Dr. Amit Prashant

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′

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Foundation Analysis and Design: Dr. Amit Prashant

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′

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Foundation Analysis and Design: Dr. Amit Prashant

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

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Foundation Analysis and Design: Dr. Amit Prashant

Determination of Effective Dimensions for Eccentrically Loaded foundations (Highter and Anders, 1985) 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

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Foundation Analysis and Design: Dr. Amit Prashant

Determination of Effective Dimensions for Eccentrically Loaded foundations (Highter and Anders, 1985) 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′

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Foundation Analysis and Design: Dr. Amit Prashant

Determination of Effective Dimensions for Eccentrically Loaded foundations (Highter and Anders, 1985)

Case V: Circular foundation

eR

R

L′ =

A′ B′

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Foundation Analysis and Design: Dr. Amit Prashant

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

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Foundation Analysis and Design: Dr. Amit Prashant

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

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Foundation Analysis and Design: Dr. Amit Prashant

Bearing Capacity of Footings on Slopes Meyerhof’s (1957) Solution Cohesive Soil

φ′ = 0

qu = c′N cq

Ns =

γH c

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Foundation Analysis and Design: Dr. Amit Prashant

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

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Foundation Analysis and Design: Dr. Amit Prashant

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

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Foundation Analysis and Design: Dr. Amit Prashant

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

For

Df B

= 0.5

45

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Foundation Analysis and Design: Dr. Amit Prashant

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

For

Df

= 1.0

B

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Foundation Analysis and Design: Dr. Amit Prashant

Bearing Capacity of Footings on Slopes 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

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Foundation Analysis and Design: Dr. Amit Prashant

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 ′

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