Formula Sheet LE1 Chap3
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Transcript of Formula Sheet LE1 Chap3
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: