3.3 Terzaghi’s Bearing Capacity Theory

qu=c 'N c+qNq+12γB N γ (continuous∨strip foundation)

qu=1.3c 'N c+qN q+0.4 γB N γ ( square foundation) qu=1.3c 'N c+qN q+0.3 γB N γ (circular foundation )where c’ = cohesion of soil γ = unit weight of soil

q = γ Df = surcharge

N cN qN γ=¿ bearing capacity factors, fcn of soil friction angle ϕ ’

For foundations that exhibit the local shear failure mode :

qu=23c'

N ' c+qN ' q+12γB N ' γ ( strip foundation)

qu=0.867c 'N ' c+qN 'q+0.4 γB N ' γ (square foundation ) qu=0.867c 'N ' c+qN 'q+0.3 γB N 'γ (circular foundation)

3.4 Factor of Safety

qall=qu

FS OR Net stress increase∈soil=

qnet (u)

FS

qnet (u )=qu−q

where qnet (u ) = net ultimate bearing capacity

q=γ Df

so, qall(net)=qu−q

FS

3.5 Modification of Bearing Capacity Eqns for Water Table Case 1: 0≤ D1≤ Df

q=D1 γ+D 2 ( γ sat−γw )=effective surcharge

where γ sat = saturated unit weight of soil

γw = unit weight of water = 9.81 kN/m3

γ in the last term of the eqn becomes γ '=γ sat−γw Case 2: 0≤d≤ B

γ in the last term of bearing cap. eqn becomes

γ=γ '+ dB

(γ−γ ' )

Case 3: When water table is located so that d≥B , no changes.

3.6 General Bearing Capacity Equation

qu=c 'N cFcsFcd Fci+qN qFqsFqd Fqi+12γB N γ F γsF γd F γi

where c’ = cohesion q = effective stress @ level of bottom of foundationγ = unit weight of soilB = width of foundation (= diameter for circular foundation)F cs ,Fqs ,F γs = shape factors

F cd , Fqd , F γd = depth factors

F ci ,Fqi ,F γi = load inclination factors

N c ,N q ,N γ = bearing capacity factors

Table 3.4 Shape, Depth and Inclination Factors

ShapeF cs=1+(BL )(N q

N c)

Fqs=1+( BL ) tan ϕ'

F γs=1−0.4( BL )Depth

Df

B≤1

For ϕ=0Fcd=1+0.4( D f

B )Fqd=1

F γd=1For ϕ '>0

F cd=Fqd−1−Fqd

N c tanϕ'

Fqd=1+2 tan ϕ' (1−sin ϕ' )2(Df

B )F γd=1

Df

B>1

For ϕ=0 F cd=1+0.4 tan−1( Df

B ) in radians

Fqd=1 F γd=1

ϕ '>0F cd=Fqd−

1−Fqd

N c tanϕ'

Fqd=1+2 tan ϕ' (1−sin ϕ' )2 tan−1(D f

B )¿ radians

F γd=1Inclination

F ci=Fqi=(1− β°ϕ ' )

2

F γ i=(1−β

ϕ' )β = inclination of the load on the foundation with respect to the veritical

3.8 Effect of Soil Compressibility

qu=c 'N cFcsFcd Fcc+qN qFqsFqd Fqc+12γB N γ F γsF γd F γc

I r=Gs

c '+q ' tan ϕ'

where I r = rigidity index at depth approx. B/2 below bottom of foundation

Gs = shear modulus of soil = E s/2(1+μ) q = effective overburden pressure at a depth of Df +B /2

I r (cr )=12 {exp [(3.30−0.45

BL )cot(45−ϕ'

2 )]}variation of I r (cr )with B/L are in table 3.6

If I r≥ I r (cr), then

F cc¿ Fqc¿F γc=1

If I r ¿ I r (cr )F γc¿ Fqc=exp¿

For ϕ=0,

F cc=0.32+0.12BL

+0.60 log I r

For ϕ '>0,

F cc¿ Fqc−1−Fqc

N q tanϕ '

Figure 3.12 Variation of F γc=Fqc with I r and ϕ '

3.9 Eccentrically Loaded Foundations

qmax, min=QBL

±6M

B2L ¿

where Q = total vertical load M = moment on the foundation

e=MQ

qmax, min=QBL

(1± 6eB

) ¿

if qmin is (-) tension foundation separation, so new

qmax=4Q

3 L(B−2e)

FS when surface strip foundation is subjected to eccentric loading

FS=Qult

Q

3.10 Ultimate Bearing Capacity under Eccentric Loading – One-Way EccentricityEffective Area Method

B’ = effective width = B-2e L’ = effective length = L

If e is in the direction of length of foundation, L’ = L-2eB’ = B

The smaller of the two dimensions is the effective width of the foundation.

Ultimate bearing capacity:

q ' u=c 'N cFcsFcd Fci+qN qFqsFqd Fqi+12γB ' N γ F γs F γdF γi

use table 3.4 for F csFqsF γs with L’ and B’

use table 3.4 for F cd FqdF γd with B

Total ultimate load that the foundation can sustain

Qult=q'u(B')(L')

where (B' ) (L' )=A ' = effective area

FS=Qult

Q

Prakash and Saran TheoryUltimate load per unit length of a continuous foundation:

Qult=B[c 'N c ( e )+q Nq (e)+12γB N γ (e)]

where N c (e) , Nq (e ) , N γ (e) = bearing capacity factors under eccentric

loadingRectangular foundations, ultimate load is given by:

Qult=BL [c 'N c (e ) Fcs (e)+q Nq (e)Fqs(e)+12γB N γ (e)F γs(e)]

where Fcs (e)Fqs (e)F γs(e) = shape factors

Also recommended by Prakash and Saran

F cs (e)=1.2−0.025LB

(with a minimum of 1.0)

Fqs(e)=1

F γs (e )=1.0+( 2eB

−0.68) BL +[0.43−( 32 )( eB )](BL )

2

3.11 Bearing Capacity – Two-way Eccentricity

eB=M y

Qult

eL=M x

Qult

Qult=q 'u A '

where q ' u=c 'N cFcsFcd Fci+qN qFqsFqd Fqi+12γB ' N γ F γs F γdF γi

A'=effectivearea=B ' L 'use table 3.4 for F csFqsF γs with L’ and B’

use table 3.4 for F cd FqdF γd with B

Case 1: eL /L≥16

and eB /B≥16

A'=12B1L1

where B1=B(1.5−3eBB

) and L1=L(1.5−3e L

L)

L’ is the larger of B1 and L1. So effective width:

B'= A 'L'

Case 2: eL /L<0.5 and 0<eB/B< 16

A'=12

(L1+L2 )BL1∧L2 from figure 3.21b:

Effective width and length:

B'= A 'L1∨L2( thelarger )

L'=L1∨L2 (the larger )

Case 3: eL /L<16

and 0<eB/B<0.5

A'=12

(B1+B2 )LEffective width and length:

B'= A 'LL'=L

B1∧B2 from figure 3.22b:

Case 4:eL /L<16

and eB /B< 16

A'=L2B+ 12

(B+B2 )(L−L2)

Effective width and length:

B'= A 'LL'=L

B2 and L2 from figure:

Case 5: Circular Foundation (under eccentric loading, e is always one way)Effective length:

L'= A 'B '

A’ and B’ from table: