Soil-Bearing Capacity for Shallow Foundations

15
Soil-Bearing Capacity for Shallow Foundations
The lowest part of a structure is generallyreferred to as the fi'rr.r
ndation. Its function
is to transfer the load of the structure to the soil on which it is resting.A properly designed foundation transfers the load throughout the soil without overstressingthe
soil. Overstressingthe soil can result in either excessivesettlemcntor shearfailure of
the soil, both of which causedamage to the structure.Thus, geotechnicaland structural engineerswho design foundertionsmust evaluatethc bearing capacity of soils.
Depending on the structure and soil encountercd.various types of foundations
are used.Figure 1-5.1
showsthe most common typcs of foundations.A spreadfooting
is simply an enlargementof a load-bearingwall or column that makes it possibleto
spread the load of thc structure over a lzrrgerarea of the soil. In soil with low loadbearing capacity,the size of thc spread lootings required is impracticably large. In
that case,it is more economicalto constructthe entire structure over a concretepad.
This is called a mot fburulation.
Pile and drilled sha.ftfoundations are used for heavier structures when great
depth is required for supporting the load. Pilesare structural members made of timber, concrete,or steel that transmit the load. of the superstructurcto the lower layers of the soil. According to how they transmit their load into the subsoil,piles can
be divided into two categories:friction piles and end-bearingpiles.In the caseof friction piles, the superstructureload is resistedby the shear stressesgenerated along
the surfaceof the pile. In the end-bearingpile, the load carried by the pile is transmitted at its tip to a firm stratum.
ln the caseof drilled shafts.a shaft is drilled into the subsoil and is then filled
with concrete.A metal casingmay be usedwhile the shaft is being drilled. The casing
may be left in place or may be withdrawn during the placing of concrete.Generally,
the diameter of a drilled shaft is much larger than that of a pile. The distinction between piles and drilled shaftsbecomeshazy at an approximatediameter of 1 m (3 ft),
and the definitions and nomenclatureare inaccurate.
Spread footings and mat foundations are generally referred to as shallow foundations, whereas pile and drilled shaft foundations are classifiedas deepfoundations.
In a more general sense, shallow foundations are foundations that have a depthof-embedment-to-width ratio of approximately less than four. When the depth-ofembedment-to-width ratio of a foundation is greater than four, it may be classified
as a deep foundation.
503
Chapter 15 Soil-Bearing Capacity for Shallow Foundations
I
V
, .
i.i,:
t:t:
t
t
i
.:a:.:
::.::j
l
,::,;
:,l'ri
"' .. ," 'j .' :l l. ,i . . :
'..'.:,:
.',,:
, l' irlr.i
'"
' i" i,' r.' '
I ;,1
:.il,,.:.1,,...,,...',1'l....t
,f ,',1,1'.'1
(a)
(D)
,]
,.ti:. .
'.'
+I
tt
.,]
I
t't t
, j ' - . ' . .
'.,'
l.l'rl',tt.
'.
'
.,,'.
'1'
. ,.,1''
+',.f
(d)
(a) sprcadlixrting;(b) mat foundation:(c) pile
Figure 15.7 Commontypcso1'foundations:
loundation;(d) clrillcdshaftI'oundatior.r
In this chapter,wc discussthe soil-bearingcapacityfor shallowfoundaticlns.As
mentioncd bcfore, for a foundation to function properly, (1) the settlement of soil
causcdby the load must be within the tolerable limit. and (2) shear failure of the soil
supporting the foundation must not occur. Compressibility of soil - consolidation
and elasticity thcory-was intnrduced in Chapter 10. This chapter introduces the
load-carrying capacity of shallow foundations based on the criteria of shear failure
in soil.
15.1
Ultimate Soil-Bearing Capacity
for Shallow Foundations
To understand the concept of the ultimate soil-bearing capacity and the mode of
shear failure in soil, let us consider the caseof a long rectangular footing of width B
located at the surfaceof a densesandlayer (or stiff soil) shown in Figure 15.2a.When
a uniformly distributed load of 4 per unit area is applied to the footing, it settles.If
the uniformly distributed load (q) is increased,the settlementof the footing gradually increases.When the value of q : q,, is reached (Figure 15.2b), bearing capacity
15.1 Ultimate Soil-Bearing Capacity for Shallow Foundations
4',,
l<-B
4r Loa<1per
>l
q
c
c/)
(b)
Figure 75.2 Ultimatcsoil-bcaring
capacityfor shallowfoundation:(a) modelfooting;
( b ) l o a ds c t t l e m e nr tc l a t i o n s h i p
f a i l u r e o c c u r s lt h e f o o t i n g u n d e r g o e sa v e r y l a r g es e t t l c m en t w i t h o u t a n y f u r t h e r
increaseof 17.The soil on one or both sideso1'thefounclationbulgcs.and the slip
surfacc
e x t e n d st o t h e g r o u n d s u r f a c eT
. h e l o a d - s e t t l c m e nrtel a t i o n s h i pi s l i k e c u r v e I s h o w n
i n F i g u r e l - 5 . 2 bI.n t h i s c e r s er ,7 ,i,s c i e f i n e da s t h e u l t i m a t e b e a r i n gc a p a c r t vo f s o i l .
T h e h c a r i n gc a p r t c i t yl a i l u r c j u s l c l c s c r i h c ci sl c : r l l c t la g e n r : r , ,i1h c t t r "
fuilurc ancl
c a n b e c x p l a i n e dw i t h r e f e r e n c et o F i g u r e l - 5 . 3 aW
. h e n t h e i o u n c l a t i o ns e t t l c su n d e r
O r i g in a l s u r f n c e
ol soil
l+1J+l
(4,
;
l+A+l
(b)
Figute 75'3 Modes of bearing capacity failure in soil: (a) general shear failure
of soil;
(b) local shear failure of soil
Chapter 15 Soil-Bearing Capacity for Shallow Foundations
the application of a load, a triangular wedge-shapedzone of soil (marked I) is pushed
down, and, in turn, it pressesthe zonesmarked II and III sidewaysand then upward.
A t t h e u l t i m a t e p r e s s u r e ,q , , , t h e s o i l p a s s e s i n t o a s t a t e o fp l a s t i c e q u i l i b r i u m a n d
failure occursby sliding.
If the footing test is conducted instead in a loose to medium dense sand, the
load-settlementrelationship is like curve II in Figure 15.2b.Beyond a certain value
of q : q',,,theload-settlementrelationship becomesa steepinclined straight line. In
this case,qj, is deflned as the ultimate bearing capacityof soil. This type of soil failure is referred to as local shear.failureand is shown in Figure 15.3b.The triangular
wedge-shapedzone (marked I) below the footing moves downward, but unlike general shear failure, the slip surfacesend somewhereinside the soil. Some signsof soil
bulging are seen,however.
15,2
Terzaghi's Ultimate Bearing Capacity Equation
In 1921,Prandtl published the resultsof his study on the penetration of hard bodies,
such as metal punches,into a softer material. Terzaghi (1943) extended the plastic
failure theory of Prandtl to evaluate thc bearing capacity of soils for shallow strip
footings.For practicalconsiderations,a long wall footing (length-to-widthratio more
than about five) may be called a strip .footing. According to Terzaghi, a foundation
may be defined as a shallow foundation if the depth D1 is less than or equal to its
width B (Figure 15.4).He also assumedthat for ultimate soil-bearingcapacity calculations,the weight of soil abovc the basc of the footing may bc replacedby a uniformsurcharge,q:yD,.
The failure mechanismassumedby Terzaghilor determining the ultimate soilbearing capacity (general shear failure) for a rough strip footing locateclat a depth
The soil wedge ,'1BJ
D1 measuredfrom the ground surfaceis shown in Figure 15.-5a.
(zone I) is an elastic zone. Both AJ and BJ make an zrngle{'with the horizontal.
Zones marked ll (A.lE and BJD) are the radial shear zones,and zones marked III
are the Rankine passivezones.The rupture linesJD and J E are arcsof a logarithmic
spiral, and DF and EG are straight lines. ,48, BD, EG, and DF make angles of
Unitweightof soil= Y
q -^{Dr
Dt< B
Figure 15.4 Shallowstrip footing
15.2 Terzaghi'sUltimate Bearing CapacityEquation
l+
l<--
B ->-l
B-2b--*tt
cb
cos Q
(b)
Figure 15.5 Terzaghi'.s
bearingcapacityanalysis
45 - 0'12 degreeswith the horizclntal.The equation of the arcs of the logarithmic
spirals J D and J E may be given as
r : r.,eor^no'
If the load per unit area,qr!i is applied to the footing and general shear failure
occurs, the passiveforce Pn is acting on each of the faces of the soil wedge A BJ. This
concept is easy to conceive of if we imagine that AJ and B.l are two walls that are
pushing the soil wedges AJEG and BJDE respectively,to cause passivefailure. p,
should be inclined at an angle 6 (which is the angle of wall friction) to the perpendicular drawn to the wedge faces(that is, ,4"/and BJ).ln this case,6 should be equal
to the angle of friction of soil, @'. BecauseAJ and BJ are inclined at an angle
@l to
the horizontal, the direction of P,, should be vertical.
Now let us consider the free body diagram of the wedge ABJ as shown in Figure 15.5b.considering the unit length of the footing, we have,for equilibrium,
: -W * 2Csine, + 2p,,
(q,,)(zb)(l)
(1s.1)
whereb : Bl2
lV : weight of soil wedge ABJ : ybz tan g'
C : cohesive force acting along each face , AJ and BJ, that is equal to
the unit cohesiontimes the length of each face : c,bl(cos $,)
Thus,
Zbq, : 2P, i 2bc' tan Q' - yb2 tan S'
(rs.2)
Chapter 15 Soil-Bearing Capacity for Shallow Foundations
s , , : + * c , t aQn, - ! t u n 6 ,
( 1s.3)
The passivepressurein Eq. (15.2)is the sum of the contributionof the weight
of soil 7, cohesionc', and surchargeq. Figure 15.6showsthe distribution of passive
pressurefrom eachof thesecomponentson the wedgefaceBJ. Thus,we can write
(15.4)
Po: ll@tanQ')2K, * c'(btang')K, + q(btanQ')K,
whereKn, K,, and K,,are earthpressurecoefficientsthat arefunctionsof the soil friction angle,@'.
T
I
I
H-btan
t _
t1
Il-
TI
I
1l nr
l )
lt
.I
(b)
TI
I
'u i- t
lry
Note:H=btan0'
t ?
er=
II
)yu2xr+
c ' ' H K , . +q H K , ,
J
(c,
Figure 15.6 Passiveforce distributionon the wedgefaceBJ shownin Figure 15.5:(a) contribution of soil weight7; (b) contributionof cohesionc'; (c) contributionof surchargeq.
15.2 Terzaghi'sUltimate Bearing CapacityEquation
509
Combining Eqs. (15.3) and (15.4),we obtain
eu:
c'N, + 4Nq+
I
;lBlV,
where
l/, : tan 0'(K, + 1)
(1s.s)
(1s.6)
(rs.7)
N,, - K,, tan $'
N, : ) tan g'(K" tan ry'' - I )
The terms N,, l/r, and N, are, respectively.the contributions of cohesion,surcharge,and unit wcight of soil to the ultimate load-bearingcapacity.It is extremely
tedious to evaluate K,, K,,,and Kr. For this reason,Terzaghi used an approximate
m c t h o d t o d e t e r m i n e t h e u l t i m a t e b e a r i n g c a p a c i t y ,q , , . T h e p r i n c i p l e so f t h i s a p p r o x i m a t i o nf o l l o w :
1 . I f c ' : 0 a n d s u r c h a r g e( q ) : 0 ( t h a t i s , D , : 0 ) . t h e n
( l s.8)
Q,,:4y:lYBN,
2 . I f y - 0 ( t h a t i s , w e i g h t l e s s o i l ) a n d r 7: 0 , t h en
(1-5.e)
4,,:4,-t'N,
3. If 7 : 0 (weightlcss oil) and c' !
0. rhen
Q,,:Q,t :7N,r
( 1 s .01)
By the method of superimposition,when the effectsof the unit weight of soil,
cohesion,and surchargeare considered,we have
Q,: e, + eq * ey : c'N, * eNq+ lyBN,
( r s .11)
Equation (15.11) is rcferred to as Terz.aghi's
bearing capucity equation. The
terms N,, {, and N, are called the bearingcqpacityfactors.The valuesof these factors are given in Table 1-5.1.
For square and circular footings, Terzaghi suggestedthe following equations
f o r u l t i m a t es o i l - b e a r i n cga p a c i l y :
T h e s q u a r eI ' o o t i n gi s
qu : 1'3c'll/, + qNn + O.4yBN^)
' ' {
(rs.12)
{
The circular footing is
qu:
1 . . 3 c ' N *, Q N , t+ A 3 y B N ,
where B : diameter of the footing.
(15.13)
510
Chapter 15 Soil-Bearing Capacity for Shallow Foundations
Tabte 15.1 Terzaghi'sBearing Capacity Factors-N,, N4 and Nr-Eqs.
(15.1
3)
a'
(deg)
0
I
2
3
,1
,5
6
7
6
9
l0
t1
12
l-)
l4
l5
t6
17
ItJ
l9
20
21
22
L-)
24
25
o'
(deg)
N"
Nq
N;
5.70
6.00
6.30
6.62
6.97
1.00
1.ti)
1.22
1.35
1.19
1.64
1.81
2.00
2.21
2.44
2.69
2.98
3.29
0.00
0.01
0.04
0.06
0.10
0.14
0.20
0.2'7
0.35
0.44
0.-56
0.69
0.85
1.04
|.26
1.52
1.82
2.18
2.59
3.07
26
27
28
29
30
31
-1.r)1+
46
4.31
-5.09
6.00
7.08
n.34
41
t.J+
7.73
u.15
8.60
9.09
9.61
10.16
10.76
11.41
12.11
12.86
13.68
14.60
15.12
16.-s6
17.69
18.92
20.27
21.75
23.36
25.t3
-t.o-J
1.02
4.45
4.92
5.45
6.04
6.70
7.44
8.26
9.19
t0.23
I L40
t2.'72
N"
27.09
29.24
31.61
,)+.1+
-)L
33
34
35
36
'-\7
3rJ
39
40
4l
42
+-)
44
A<
4u
49
-50
3'7.t6
40.41
44.04
4[t.09
52.64
57;75
63.53
70.01
77.50
u5.97
95.66
I 0 6 . 8I
t19.67
l34.5tt
l5l .9-s
172.28
196.22
224.55
25t3.2u
298.11
347..50
(15.11),(15.12),and
Nq
14.21
15.90
17.8I
l9.gti
22.46
25.28
28.52
32.23
36.-s0
41.44
47.16
53.frO
6l .-5,5
10.61
81.2'7
93.85
108.7-5
126.50
147.74
173.28
204.19
241.r30
2U7.r3-5
344.63
4 t- 5 . 1 4
N;
9.U4
11.60
13.70
16.18
19.13
22.65
26.87
31.94
38.04
45.41
54.36
65.27
78.61
95.03
I15.31
140.-5
I
l7|.91)
2fi.56
261.60
325.34
407.tl
-s12.t34
650.61
ti3t .99
1072.80
" From Kumbhoikar(1993)
E,quation(15.1I ) was derived on the assumptionthat the bearing capacityfailure of soil takes place by general shear failure. ln the caseof local shear failure, we
may assumethat
t
-
)
t
3L
(1 s . 1 4 )
and
hn$' : \tang'
(1s.1s)
The ultimate bearingcapacityof soil for a strip footing may be givenby
q , , : Z ' N L+ S N L+ I Y B N ,
(1s.16)
15.2 Terzaghi'sUltimate Bearing Capacity Equation
511
Table 15'2 Terzaghi's
ModifiedBearingcapacityFactors-N:, N;,and Ni-Eqs. (15.16),
( 1 s . 1 7 a) ,n d( 1 5 . 1 8 )
o'
(deg)
0
1
2
3
4
5
o
7
IJ
9
10
ll
t2
13
t4
l-5
l6
I'7
Iu
l9
20
21
22
L-)
a/1
25
N"
-5.7t)
5.90
6.10
6.30
6.51
6.74
6.91
7.22
7.47
7.14
u.02
I.i.32
u.63
ti.96
9.31
9.61
I0.06
10.41
r0.90
I 1.36
I l.u5
12.37
12.92
13.51
t4.14
14.u0
Nq
1.00
1.07
1.14
1.22
1.30
1.39
1.49
1.59
I.l0
l.u2
t.94
2.0r1
2.22
2.3lJ
2.55
2.92
3 .l 3
3.36
3.61
3.utt
4.tl
4.48
4.82
5.20
5.60
N,
0.00
0.00.5
0.02
0.04
0.0.55
0.074
0.10
0.l2ti
0 .l 6
0.20
0.21
0.30
0.3.5
0.12
0.4n
0.-57
0.6'l
o.'16
().t3u
1.03
t.t2
1.3.5
l.-55
|.74
t.97
o'
(deg)
26
2"/
28
29
30
31
-)L
33
34
35
36
37
38
39
40
4l
A'
N"
15.53
16.30
17.13
r 8.03
I t3.99
20.03
2t.16
22.39
23.72
25.r8
26.77
213.51
30.43
32.53
34.n7
37.45
40.33
+-ll
4-1.-\4
44
47.13
-51.17
-55.73
60.91
66.u0
73.-55
til.31
4.)
16
47
4u
19
50
Nq
6.0-5
6.54
7.07
7.66
8.31
9.03
9.82
10.69
1.67
12.75
t3.97
r5 . 3 2
r6.u-5
113..56
20..s0
22.70
25.2t
213.06
31.34
3 5 . 1r
39.48
44.54
-50.46
57.41
6-5.60
ry
2.59
2.88
3.29
3.76
4.39
4.it3
.5.51
6.32
7.22
it.3-5
9.41
10.90
l 2 ; 75
14.71
n.22
19.75
22.50
26.25
30.40
36.00
4t.70
49.30
59.25
71.45
u5.75
l.tJ
The modified bearingcapacityfactorsN i , N i,,and Ni are calculateclby usingthe same
r(] tan
g e n e r a le q u a t i o na s t h a t f o r N , , N , , . a n d
{ , b u t h y s u b s t i t u t i n gd , : t a n
@,)
for $' . The valuesof the bearing capacityfactorsfor a local shearfailure are givenin
Table 15.2.The ultimate soil-bearingcapacityfor squarcand circular footings for the
local shear failure case may now be given as follows
[simitar to Eqs. (15.12) and
( 1 s.1 3 )l :
T h e s q u a r el o o t i n g i s
eL : l.3c' N',+ qN'q+ 0.4yBN;X
(1s.
r7)
qL: L.3c'NL
+ qNL+ A3yBN
"
(1s.18)
The circular footing is
512
Chapter 15 Soil-Bearing Capacity for Shallow Foundations
For undrained condition with d : 0 and rf : cu, the bearing capacity factors
a r e N " - N ' y : 0 a n d . f y ' ,-, n J ' ,: ,1 . A l s o , l / , : l / i : 5 . 7 ' I n t h a t c a s e ,E ' q s '( 1 5 ' 1 1 ) ,
(15.12),and (1-5.13)(which are the casesfor general shear failure) take the forms
q , , : 5 ' 7 c , I, q
(strip footing)
( l 5 . 18 a )
and
q , , : ( 1 . 3 ) ( 5 . 7 ) c ,+, q : 7 . 4 1 c ,i, q
(squareand circular footing)
(1.5.1frb)
) ,h i c h a r e f o r t h e c a s eo f l o c a l
I n a s i m i l a rm a n n e r ,E q s . ( 1 , 5 . 1 6 () .1 - 5 . 1 7a) ,n d ( 1 - 5 . 1 8w
shear failure. will take the forms
, t ' , , :( ] . , , ) 1 r . t yl q : 3 . 8 c , ,
\-r
(stripfooting)
lt1
(l-s.l9a)
/
and
(lu -
* q - 4.94c,,r
tr:r(3',)r."r
t1
(squareand circular footing)
( I s . 1e b )
15.3
General Bearing Capacity Equation
After the devclopment ol"l-crzaghihbearing capacityequation, scvcral investigators
w o r k e d i n t h i s a r e aa n c lr e fi n c d t h e s o l u t i o n( t h a t i s , M e y e r h o f ,l 9 - 5I , 1 9 6 3 ;L u n d g r c n
; a l l a , 1 9 6 2 ) .D i f f e r e n t s o l u t i o n ss h o w t h a t t h c b e a r i n gc a p a c a n d M o r t e n s e n ,l 9 - 5 3 B
the vality factors N, and N,, do nttt changemuch. Howevcr, for a given valuc of <75',
ucs of N" obtaincd by different invcstigatorsvary widcly. This dilTercnceis because
of the variation of the assumptionof the wedgc shapeof soil locatcd directly bclow
the footing, as explained in thc following paragraph.
Whilc deriving the bearing capacitycquation for a strip l'ooting,Terzaghi uscd
the caseof a rough footing and assumedthat the sidesA./ and B"/ of the soil wedge
make an angle @' with the horizontal. Later model tests(for
ABJ (seeFigure 1-5.5a)
example,DeBeer and Vesic,1958)showedthat Terzaghi'sassumptionof the general
nature of the rupture surfacein soil for bearing capacityfailure is correct. However,
tests have shown that the sides ,4J and B.l of thc soil wedge AB.l make angles of
about 4-5+ 0' 12degrees,insteadof ry''.with the horizontal.This type of failure mechanism is shown in Figure 1-5.7.It consistsof a Rankine activezone A BJ (zone I), two
radial shear zones(zonesII), and two Rankine passivezones.(zonesllI). The curves
J D and J E are arcs of a logarithmic spiral.
On the basisof this type of failure mechanism,the ultimate bearing capacityof
a strip footing may be evaluatedby the approximatemethod of superimpositiondescribed in Section 1,5.2as
Q u : Q r t Q q t Q ,
( rs.20)
w h e r ee , , Q q , a n d q r a r e t h e c o n t r i b u t i o n s o f c o h e s i o n , s u r c h a r g e , a n d u n i t w e i g h t o f
soil, respectively.
15.3 General Bearing CapacityEquation
513
F<__B+l
45 - q'/2 ^<
- q'n 4s- q'tz
Figure 15.7 Soil-bearing
- generalshcarfailure
capacityc:rlculation
Reissner(1924) expresscdr/./as
t1, - ql\'1,,
( l - s . 2)1
where
gr tan4'
Nn :
d ')\
tun'(+s * T
(ls.22\
P r a n d t l ( 1 9 2 1 )s h o w e dr h a r
q, : (' ll,
( 1.5.23)
where
7\y'- (li, - l)cot g'
t
Eq (1s.22)
(ls.24)
Mcyerhof ( 1963)cxprcssedr7"as
qt : :BYlvy
(r-5.2,s
)
where
Nr: (N, * 1)tan(1.4@')
t
Eq.(1s.22)
CombiningEqs.(15.20).
(15.21),
(15.23),
and(15.26).
we obtain
-- ('N, t
t,yBl,,t,
4,,
4N,r-
(Ls.26)
(rs.27)
This equation is in the samegeneralform as that givenby Terzaghi
[Eq. (15.11)];however, the values of the bearing capacity factors are not the same.The values of N-.
{,andl/,,definedbyEqs. (15.22),(15.24),ancl(15.26),aregiveninTabtes15.3anh
514
Chapter 15 Soil-Bearing Capacity for Shallow Foundations
Table 15.3 BearingCapacityFactorsN,.,and N, [Eqr. (15.22)and (15.2a)]
a'
(deg)
U
I
2
3
A
5
6
7
6
9
l0
l l
l2
l3
l4
l-5
t6
t1
llJ
t9
20
2l
22
L-)
..,
A
25
ru"
N"
,5.14
5.38
5.63
-5.90
6.19
6.49
6 . nI
7.16
7.53
7.92
8.3-s
8.80
9.2u
9 . Ul
10.37
l0.gtJ
I 1.63
t2.34
I3.t0
13.93
14.u3
l-5.82
16.8u
18.0-5
t9.32
20.72
1.00
1.09
1.20
I.31
1.43
|.57
1.72
l.uu
2.06
l.l)
1 1 4
2.71
2.97
3.26
3.-59
3.94
4.34
4.17
5.26
5.itO
6.40
7.07
7.82
t3.66
9.60
10.66
a'
(deg)
26
27
28
29
30
31
32
33
34
3-5
36
31
38
39
40
4l
12
43
41
46
47
4u
49
50
N"
22.25
23.94
2-5.80
2'7.86
30.14
32.67
35.49
38.64
42.16
46.12
50.59
5-5.63
61.3-5
67.8'7
75.31
t33.t36
93.7|
I 0 5 . 1I
lln.37
l33.tit3
l-52.I0
113.64
t99.26
229.93
266.89
Nq
11.85
13.20
14.72
16.44
18.40
20.63
23.18
26.09
29.44
33.30
3'7.75
42.92
413.93
-55.96
64.20
73.90
u-5.38
()().02
ll -5.31
134.u8
t5u.5l
t8'7.21
222.31
265.51
3t9.07
l-5.4,but for all practicalpurposcs,Terzaghi'sbearing capacityfactorswill yield good
rcsults.Differencesin bearing capacityfactors nre usually minor compared with the
unknown soil parzrmeters.
The soil-bearingcapacity equation for a strip footing given by Eq. (1-s.27)can
be modilied for general use by incorporating the following factors:
l. depth.factor:to account for the shearingresistancedcvelopedalong the failure
surfacein soil above the base of the footingl
2. shape.factor: to determine the bearing capacity of rectangular and circular
footings; and
3. inclination factor: to determine the bearing capacity of a footing on which the
direction of load application is inclined at a certain angle to the vertical.
Thus, the modified generalultimate bearingcapacityequation can be written as
4,,:
c ' t r , , i , , i t r , , 1IV , q A , , , A r , t A , 1+i NI ,A, , A r a l r ; y B N ,
where I,.,, I.r", and ),r" : shape factors
).,.,1,
and ,\r,r : depth factors
A.q,t,
L,i, troi,and .trnt: inclination factors
(1s.28)
15.3 General Bearing CapacityEquation
515
Table 15.4 BearingCapacityFactorN, [Eq. (15.26)J
o'
(deg)
{.)
I
2
3
4
5
6
7
t5
9
l0
ll
12
l3
I4
l-s
l6
t7
Itt
l9
20
2T
22
L-)
',,1
25
26
o'
(deg)
Ny
0.000
0.002
0 . 0 01
0.023
0.042
0.070
0.106
0 . 15 2
0.209
0.280
0.361
0.47|
0.-596
0.711
0.921
t.t29
1.37-s
1.664
2.(X)3
2.403
2.87|
3.421
4.066
4.824
5.716
6.765
8.002
Ny
L).163
I1.190
13.236
l -5.66t3
r8.564
22.022
26.t66
31.14-5
37.152
14.426
53.27o
64.073
77.332
93.690
I t3.9U-5
I39.316
nt.I4l
2ll.1Ut
262.739
328.72E
4t1.322
526.444
674.90lt
u73.lt'13
l r43.934
I 5 16.0-s
I
2037.258
2'/
28
29
30
3l
-)L
33
-)+
3.5
36
3tt
39
40
4l
/a
43
44
45
46
41
48
1r)
-50
5t
)l
-s3
The approximate valucsof theseshape,clepth,and inclinzrtionfactorsrecommendecl
by Meyerhof are given in Tablc 15.-5.
F o r u n d r a i n e dc o n d i t i o n ,i f t h c f o o t i n g i s s u b j c c t e dt o v e r t i c a lt o a d i n g( t h a t
is.
a : 0"). then
o:0
L : C , ,
0
A / -
1
I
I
-
I
-
,\y; :
I
S o E q . ( 1 5 . 2 8 )l r a n s f o r m st o
.,'(i)]['
.',(?)]
q,,: S.t+r',,1
|
L
+ q
(1s.2e)
516
Chapter 15 Soil-Bearing Capacity for Shallow Foundations
Table 15.5 Mcyerhof's Shape,Dcpth. and Inclination
Factorsfor a Rectansul:rrFootins"
Shape factors
0"'
Rtr Q:
/n\
^,,-l+0.2(r/
iu,:l
ir'-l
Flr tlt' > 10":
) . , ,- I +
o r ( i ) t ^* ."+' ()
-r+or(f),^"'(rr-'5)
Depth factors
['.1171fi :
0":
/ Dr\
- I * 0.2t
I
,tr,,7
A,,,r- lyr - |
litr (i > 10":
,'\,,1-
I 1-
\ u /
.+)'+)
-, , o,('1i)'"
,
t D,\
/
,
,
"'\,
/t'rrt(+s
" (r'
lnclination
factors
0,,-
/
ft
I
^,,,:
\'
/
1",-(t
,r" \r
eo")
u
\l
.ni,/
,r" \r
a^')
" R : w i d t h o l ' l o o t i n g ; l - - l c n S t ho l l i r o t i n g
15.4
Effect of Groundwater Table
I n d e v c l o p i n gt h e b e a r i n gc r p a c i t y e q u a t i o n sg i v e ni n t h e p r e c c d i n gs e c t i o n sw e a s sumcd that the groundwater table is locatcd at a depth much greater than the width,
rB of the footing. Howcver, if the groundwater table is close to the footing, some
c h a n g e st r r e r e q u i r c d i n t h e s e c o n da n d t h i r d t e r m s o f E q s . ( l 5 . l l ) t o ( 1 5 . f 3 ) , E q s .
( l - 5 . 1 6 )t o ( 1 - 5 . 1 8 )a,n d E q . 1 5 . 2 8T
. h r e c d i f f e r e n tc o n d i t i o n sc a n a r i s er e g a r d i n gt h e
location of the groundwater table with respectto the bottom of the foundation. They
arc shown in Figure 15.8.Each of thesc conditions is briefly describednext.
o
CaseI (Figure 15.8a):lf the groundwater table is located at a distanceD above
the bottom of the foundation, the magnitude of r7in the secondterm of the
bearing capacityequation should be calculatedas
q-y(Dr-D)+y'D
( 1s.30)
15.5 Factor of Safety
517
I
v
Groundwater
table
Yrrt
(4,
I
Y
i::t'i,,
A
|
I
I
1'..r,,u1
ili,iirl.
,,,,,t',':.,
i : t , l :l tl
l:ll:
ill'1''
v
Groundwatel
',,*.r.,,,r,,,.,"r{&,t,rn',,'1,.-"'*,,,,r,,,,r,,'.,",,,.,.,r,,,,!,.,,,.
table
Y.ur
(c)
Figure 15.8 Effect of thc location of groundwater table on thc bearing capacityof shallow
fbundations:(a) Casc I; (b) Casc II; (c) Case III
where 7' : l,u, - 7,,,: effectivc unit weight of soil. AIso, the unit weight of
soil, 7, that appearsin the third term of the bcaring capacityequationsshould
be replaced by 7'.
CaseII (Figure 15.8b):lf the groundwater tablc coincideswith the bottom of
the foundation, the magnitude of q is equal to yD,, However, the unit weight, y,
in the third term of the bearing capacityequationsshould be replaced by
7,.
CaseIII (Figure l5.Bc): when the groundwater table is at a depih D below the
bottom of the foundation, q : TDt The magnitude of y in the third term of the
bearing capacityequationsshould be replacedby 7,,,.
l _
f , , : u l t D + y ' ( B- D ) l
You=T
15.5
(forD>B)
(for D < B)
(1s.31)
s*
s;i
(1s.32)
Factor of Safety
Generally, a factor of safety, F", of about 3 or more is appliedto the ultimate soilb e a r i n gc a p a c i t yt o a r r j v e a t the valueof the allowablebearingcapacity.An
of 3
{
518
Chapter 15 Soil-Bearing Capacity for Shallow Foundations
or more is not consideredtoo conservative.In nature, soilsare neither homogeneous
nor isotropic.Much uncertainty is involved in evaluatingthe basicshearstrengthparameters of soil.
There are two basic definitions of the allowable bearing capacity of shallow
foundations. They are gross allowable bearing capacity,and net allowable bearing
capacity.
The gross allowable bearing capacity can be calculated as
Q"
Q.rtt:
( 1s.33)
7,
A s d e f i n e db y E q . ( 1 5 . 3 3 ) q n l l i st h e a l l o w a b l el o a d p e r u n i t a r e a t o w h i c h t h e
soil under the foundation should be subjectedto avoid any chanceof bearing capacity failurc. It includes the contribution (Figure 15.9) of (a) the dead and live loads
(b) the self-weightof the foundation, W',; and
above the ground surface, W1p+r.1',
(c) the weight of thc soil located immediately above foundation, W5. Thus,
, - :? , : l
W1o.t,1+ WI. + W ,l t
I
ln
(1 s . 3 4 )
whcre A - areaof the foundation.
The nct allowabLehearing capacity is the allowable load per unit area of the
foundation in excessof the existing vertical effectivc stressat thc level of the foundation. The vertical effectivestressat the foundation level is equal to q - yDr. So the
n c t u l t i m a t el o a d i s
( 15.3.5a)
*Qu*4
4a(net)
Hence.
:
4rrr(nct)
4rr(nct)
:
f,
Figure 75.9 Contributions to qo11
4,,-Q
F
(ls.3sb)
15.5 Factorof Safety
519
If we assume that the weight of the soil and the weight of the concrete
trom which
the foundation is made are approximately the same,then
ws+w,_
^
Q : l D t =
A
Hence.
Wg.,*t1
r1all(nct):
:
Q,,- Q
( 1.5.-r6)
F,
A
The plan of a 4-ft-squarefooting is shownin Figure15.10.Determinethe gross
allowableload,Qun(Quu: qaux areaof the fociing) that the footingcancarry.
A
factor of safetyof 3 is needed.UseTerzaghi'sequationand assumegeneralshear
failure in soil.
:e.!.:i:iiiti:1.:t:1ia.:.r:1:t:i:.::
i::;
Y= I l0 lb/ft3
0 ' =2 0 "
r" = 200lb/ft2
Figure 15.10
Solution
Assuminggeneralshearfailure of soil,we have
q,,: I.3c'N,.* QNq+ 0.4yBN,
[Eq. (15.t2)]
:
:
From Table15.1,for 6'
20o,N, 17.69,
N,, = 7.44,andN" = 3.64,
q : rDr: ll0 x 3 : 3301b/fr2
So
q, = (1.3)(200X17.6e)
+ (330X7.44)
+ (0.4xtr10)(4x3.64)
= 4599+ 2455+ 641.:7695lblft2
{
i
Q,n:
:Qu
r,:
7695
,
= 2 5 6 5l b l f t 2
Hence,
Quy= 2565x 82 : 2565x 16 : 4L.040lb
Chapter 15 Soil-Bearing Capacity for Shallow Foundations
Example15.2
Redo ExampleProblem 15.1assuminglocal shearfailure in soil. Use Eq. L5.17.
Solution
F r o mE q . ( 1 5 . 1 7 ) .
e',,: 1.3t'Nl * eN q + 0.4yBN',
and
a
v' :
;(loq:
133.3lb/ft2
From Table15.2,N', = 11.85,Ni - 3.88,and Ni, : 1.12.So,
q ' : ( 1 . 3 ) ( 1 3 3 . 3 X 1 1 .+8 s( 1) 1 0x 3 ) 3 . 8 8+ 0 ' 4 ( 1 1 0 X 4 X 1 . 1 2 )
:2054 + 1280* 197= 3531lb/ftz
and
q!,
aat:i
3531 : r l 7 7 t b t f t ?
Hence,
Q,,x: ll77 x 87 : 1177x 16 = 18,832lb
Example15.3
A squarefooting is shownin Figure 15.11.The footing will carry a grossmassof
30,000kg. Using a factor of safetyof 3, determinethe sizeof the footing * that is,
the sizeof B. Use Eq. (15.12).
p = 1850kgim3
h'=
150
c'= 0
|<-B
_*--**r{
Figure15.11
15.5 Factor of Safety
521
Solution
n;;;t"* tharsoildensity: 18s0kg/m3.So
1850x 9.81
?: ff:
18.1skN/m3
Total grossload to be supportedby the footing is
(30.000)9.81
: 294.3kN = O"u
100,
From Eq. (I5.I2)
qu: 7.3t'N,* QNq+ 0.4yBN,
With a factor of safetyof 3
a^,: \:J
+ o.4yBNr)
l{t.:.'ru.+ QNq
(a)
294.3
E
(b)
J
Also,
Qan
Qan:V:
From Eqs. (a) and (b),
e#
6'
:
+ QNq
+ o.AyBN,)
{1t.r.'t
J'
(c)
From Table 15.1",
for Q' = 35",N,: 57.75,N,, * 41..44,
and N, * 45.41.Substituting thesevaluesinto Eq. (c) yields
294.3:;L(1.3X0Xs7.7s)
1-
.2
b'
J-
+ (18.1s
x r)(41.44)
+ 0.4(1s.15)(8X45.41)l
or
*
294.3
: 250.1 + 109.9
ur?,
s
1
The preceding
equationmaynowbesolvedby trial anderror.andfrom thatweget
B - 0.95m
r
Chapter 15 Soil-Bearing Capacity for Shallow Foundations
Example 15.t[
with an F" = 3
Refer to Example 15.1.Determine the net allowableload Oal(net)
againstthe net ultimatebearingcapacity.
Solution
From Examplel5.l
q,,- 7695tblf{
= Qu* 4 : "1695* 330 : 7365lblft2
{a(net)
?455lblft2
4a'(ner):
ry: ry:
So
: (2455X4')= 39'280lb
: (4orr(n",lXB'?)
Oarr(ner)
I
Example15.5
Determinethesafegrossload(factor
A squarefootingis shownin Figure15.12.
of safetyof 3) thatthefootingcancarry.UseEq.(15.28).
I
I
v . . .
Y =1 6k N / m J
i.:
c'=0
'
'
Q ' =3 2 '
i
A I
^-1
o ' 5m
CroundwaterIable
I
Y
',,y,',u,,:.:{*.,ru,.:iry**i1tit
i :"r.,1
=
:'.lr:,1 %ar r9.5kN/m3
.. ]
0.5m
l
r
;ilmrfft$q*rult*g:r;r;1l
.
'
I F{-_.-
,
'
i
l
l'2 m---------N
Figure rs.r2
Solution
FromEq. (15.28).
qu = c'tr".tr'4N,
* qtrn,troa{+ j7',\r.n ydBN,
(Note:)\r1,trqr,and iri are all equalto 1 becausethe load is vertical.)
Because
c' : 0.
"{
nu: q)r,,,\pN,+ j7',\r,n78N,
:
15.5,
FromTables15.3and 15.4,for$' :32",ltln: 23.18andN" : tL.AL.FromTable
^ .,1/ (B7 \/ t.u,n/..1-+* 5o ' \
, \ n:,, \ r , : 1 + 0
T )
/ttt
: 1 + 0 . 1(
/
ar\
: 1.32s
+,
rr,),""'[+s
)
15.6 UltimateLoadfor ShallowFoundationsunderEccentricLoad
523
/ Dr\
/
6'\
) t q 4 : t r y "=t I + 0 ' l [ ;
45 + a 1
J
t
a
n
[
z /
\
' " :
'
r
r
\
/ r \
/
: I + 0.1(
: t.ts
, , ) t a n ( + s+ ; )
\t.2./
z /
\
The groundwater
tableis locatedabovethe bottomof the foundation,
so,from
Eq.(15.30),
- 9.81):12.845kN/mz
q : (0.sX16)+ (0.5)(19.5
Thus,
,.
(Ix1e
s e81
15)(1,
2)(22
02)
x132sx1
:_r,y':lr5?f:i:li
648.8= 2 1 6 ' 3 k N / m z
e,
Qat:i =;
Hence.the srossload is asfollows:
: 311.5kN
Q : q^u(B'): 216.3(1.2)2
15.6
I
Ultimate Load for Shallow Foundations
under Eccentric Load
One-Way Eccentricity
To calculate the bearing capacity of shallow foundations with eccentric loading,
Meyerhof (1953) introduced the concept of e.ffective
area.This concept can be explained with referenceto Figure 15.13,in which a footing of length L and width B is
subjectedto an eccentric load, Q,. If Q, is the ultimate load on the footing. it may
be approximated as follows:
l. Referring to Figures 15.13band l5.l3c, calculatcthe effectivedimensionsof
the foundation. If the eccentricity(e) is in the r direction (Figure 15.13b),the
e.ffect iv e dime nsio ns ar e
X : B - 2 C
and
Y _ L
However, if the eccentricityis in the y direction (Figure 15.13c),the effective
d i m e n s i o n sa r e
Y:L
Z:
and
X--B
2. The lower of the two effective dimensions calculated in step 1 is the eJfective
width (B') and the other is the effectivelength (L').Thu1
B' - X or I whichever is smaller
L' : X or )j whichever is larger
524
Chapter 15 Soil-Bearing Capacity for Shallow Foundations
,I
i
D,
al
I
I
t
i
,
Q
u
, e I
i<+l
i
i
.
:
+
' :
I
:
Bxa
I
(a)sJctic,n
A.)'
t+B--'|+r
N
I
*l
\]
L
\
tl
*]
T
l-l
X=l]-2c=ll'
X=l]=R'
(b) Plan
(c) Plan
Figure 75.13 Llltirnateloadfor shallowfoundationundereccentricload
3. So the effcctive area is cqual to ^B'times L'. Now. using the effectivewidth, we
c a n r e w r i t e E q . ( 1 5 . 2 t t )a s
+ ],1r.'l,yayB',
N,
qu : c' Ic"Ic,1N,* qA,n")tnaNn
(1s.37)
Note that the preceding equation is obtained by substitutingB' for B in
While computing the shapeand depth factors,one should use
Eq. (1-5.28).
B' for B and L' for L.
4. Once the value of 4,,is calculatedfrom Eq. (15.37),we can obtain the total
srossultimate load as follows:
Q,,: q,,(B'L')- quA'
(1s.38)
where,4' : effectivearea.
Two-Way Eccentricity
When foundations are subjected to loads with two-way eccentricity, as shown in Figure 15.14,the effectivearea is determined such that the centroid coincideswith the
load. The procedure for flnding the effective dimensions, B' and L' , are beyond the
scope of this text and readers may refer to Das (1,999).Once B' and L' are determined, Eqs. (15.37)and (15.38)may be used to determine the ultimate load.
15'6 ultimate Load for shailow Foundations under Eccentric Load
t
-r-l I
A
1
l
l
l
it l r
t*
(a)
o-
,
Figure 75.74 Foundation subjcctcd to two-wiiy ccccntr.icity
E xa mp l e1 5 .6
A rectangular
footing1.-5
m x 1 m is shownin Figure15.15.Determinethe magnitudeof thegrossultimateloadappliedeccentriiallyfor bearingcapacity
lailuie
in soil.
1
4,,
=
i*l
" tt.L',
I tlt
I
t +
l
J",
II
l . - 5n t
I
I
Y= lil kN/m1
c'=0
0 ' =3 0 '
I
t).I nr i*|
J
----*-ir--*--------
l+lnt+1
i
Figure 15.15
526
Chapter 15 Soil-Bearing Capacity for Shallow Foundations
Solution
F r o mF i g u r el 5 . l 3 ba n d 1 5 . 1 5 ,
X : B - 2e : 1.* 2e * 1 * (2X0.1): 0.8m
Y: L:1..5m
Str the effectivewidth (B') - 0.8 m and the effectivelength (L') - 1.5m. From
Eq. (15.37),
+ ii7"Iy,lf B'N,
q,, - qLrstrqaN,/
FromTable15'5,
FromTables15.3and75.4,for$' : 30',l/s: 18.4andN, : 15.668'
-t
i u . :i , , - I + o . r ( # ) , , " ' ( o
5)
- I + o 1/(oi .. ;s/\, " ""' /( 4 sI 3 0 \ : i l 6
; )
- Ay,:I + o.'(?),""(-t.
tr,,,
5)
/ r \
/
: r +0.1(*/,,"(os *
10\
i )
: 1.2t7
So
4u: (1 x 18X1.16)(1.211)(18.4)
+ ( 1 x 1 . 1 6 x 1 . 2 1 7 X 1 8 ) ( 0 . 8 X 1 s -. 6 6 2
87
)k N / m 2
Hence,from Eq. (15.38),
* 752kN
Q,: qu(B'L') : (627)(0.8X1.5)
15.7
Bearing Capacity of Sand Basedon Settlement
Obtaining undisturbedspecimensof cohesionlesssand during a soil exploration program is usually difficult. For this reason, the results of standard penetration tests
(SPTs) performed during subsurfaceexploration are commonly used to predict the
allowablesoil-bearingcapacityof foundations on sand.(The procedure for conducting SPTs is discussedin detail in Chapter 17.)
Meyerhof ( 1956)proposed a correlation for the net allowable heuringpressure
for foundations with the corrected standardpenetration resistance,N.,,,.The net allowable pressurewas defined in Eq. (15.36).
(For definition of the corrected standard penetration resistance,please see
S e c t i o n1 7 . 5 . )
15.7 Bearing Capacity of Sand Based on Settlement
527
According to Meyerhof's theory, tor 25 mm (1 in.) of estimated maximum
settlement.
: 11.981/.,,,
qnu(n"r)(kN/mr)
(for B = 1.22m)
1\2
: z.lulv..
fnuln"ty(kN/m'];
,,(+:+
l
'
2
f
l
B
\
)
( 1s.3e)
$"tB>l'22m)
(1s.40)
where N.u, : corrected standardpenetration number
N o t e t h a t i n E q s . ( 1 , 5 . 3 9a) n d ( 1 - 5 . 4 0B) i s i n m e r c r s .
In English units.
:
4urr1n"ry(kip/f,')
+
(turB < 4 fr)
(r-5.4
)1
and
q u r r 1 u " r ; ( k i-p /{fot u
, )( ' + , ) '
( f o r B> 4 i r )
( r.5.42)
Since Meyerhof proposed his original correlation, researchershavc obscrvcd
that its resultsare rather conservative.Latcr, Mcyerhof ( 196-5)
suggcstedthat the nct
a l l o w a b l eb e a r i n gp r e s s u r cs h o u l d b c i n c r e a s c db y a b o u t - 5 0 , / oB. o w l e s ( 1 9 7 7 )p r o posed that the modified form of the bearing pressureequations bc exprcsscclns
: / tmq'.)r o r . n , ( * )
r7"111u",;(kN
,nx s -< 1.22m)
(1s.43)
and
: t 1.981/.,,,
r7u111,,",y(kN/mt)
(f4ryf)'
.,(*)
( l o r r 9> 1 . 2 2n )
( 1.5.44)
where {7 : dcpth factor - | + 0.33(DtlB) - I
S , .: t o l e r a b l ee l a s t i cs e t t l e m e n t i,n m m
(1.5.4-5)
-) -l
Again, the unit of B is meters.
I n E n g l i s hu n i t s ,
:
r7n111n",;(kip/frr)
*4,r"
(for B < 4 ft)
( 1s.46)
and
(kiPift2) :
4"rr1".r;
N,",(B+1\t
4 \
B
)
F,tS"
(forB>4ft)
(rs.41)
where {, is given by Eq. (15.45)
S" : tolerable elasticsettlement,in in.
The empirical relationsjust presentedmay raisesome questions.For example,
which
value of the standardpenetration number shouldbe used,and what is the effect
of the
Chapter 15 Soil-Bearing Capacity for Shallow Foundations
water table on the net allowablebearing capacity?The designvalue of ly'.,,,should be
determined by taking into accountthe N.,,,valuesfor a depth of 2B to 38, measured
from the bottom of the foundation. Many engincersare also of the opinion that the
N.,,.value should be reduced somewhatif the water table is close to the foundation.
However, the author believesthat this reduction is not required becausethe penetration resistanccreflectsthe location of the water table.
15.8
Plate Load Test
In sttmc cases.conducting licld load tests to determine the soil-bearingcapacity clf
foundations is dcsirable.Thc standard method for a field load test is given by the
A m e r i c a n S o c i e t y f o r T c s t i n g a n d M a t e r i a l s ( A S T M ) u n d e r D e s i g n a t i o nD - 1 1 9 4
( A S T M , 1 9 9 7 ) .C i r c u l a r s t e e l b e a r i n gp l a t e s l - 5 2t o 7 6 0 m m ( 6 t o 3 0 i n . ) i n d i a m e t e r
and 30-5mm X 305 mm ( I ft X I ft) square platcs are used for this type of test.
A d i a g r a m c l l ' t h e l o a d t c s t i s s h o w n i n F i g u r c l - 5 . 1 6T. o c o n d u c tt h e t e s t ,o n e
must havea pit of dcpth l)1 excavated.The width of the test pit shoulclbe at lcast four
t i m e st h e w i d t h o f t h e b e a r i n gp l a t et o b e u s e d1 o rt h c t e s t .T h e b c a r i n gp l a t c i s p l a c e d
on thc soil at the botton.r<lf the pit, and an incremcnterlload on the bearing plate is
a p p l i c d . A f t e r t h e a p p l i c a t i o no f a n i n c r e m e n t a ll o a d , e n o u g h t i m e i s a l l o w e d f o r
s e t t l e m e n tt o o c c u r .W h c n t h e s e t t l c m e n to f t h e b e a r i n gp l a t e b e c o m c sn e g l i g i b l c ,
a n o t h e r i n c r c m e n t a ll o a d i s a p p l i e d .I n t h i s m a n n e r ,a l o a d - s e t t l e m c npt l o t c a n b e
o b t a i n e d ,a s s h o w n i n F i g u r e l - 5 . 1 7 .
From the rcsults of ficld load tests,the ultimate soil-bearingcapacityof actual
footings can be approximatcd as iollows:
For clays,
'/rr(liuting) :
( 1-s.4rJ)
4r,1plate)
Reaction beant
Load pcr unit area,q
$$FW€S*SY$$$S!
Jack
.t;
Dl
:l
ri
;
tr
a
Figure 15.16 Diagram of plate load tesl
Figure 15.17 Typicalload-settlement
curve
obtainedfrom platcloadtest
15.8 Plate Load Test
For sandy soils,
Blroottng;
4u(footing)
:
Qu@late)
B(ptate)
(1s.49)
For a given intensity of load q, the settlement of the actual footing can also be
approximated from the following equations:
In clay,
: S",n'",.,ffi
Se(footing)
(1s.,s0)
: t"*,",",I
sc('x,ting)
O#lffin*]'
(r s . s r )
l n s a n d ys o i l ,
Housel (1929)alsoproposeda method for obtaining the soil-bearingcapacityof
a footing that rests on a cohesivesoil for a given settlement S,..According to this procedure,the total load carried by a footing of area A and perimeter p can be given by
Q:Aq+Ps
( l s.s2)
where q : compressionstressbelow the footing
s : unit shear stressat the perimeter
Note that q and s are the two unknowns that must be determincd from the results of the {ield load testsconducted on two different-sizeplates. If e, and e2are
the loads required to produce a settlements. in plates I and2, respectively,then
Qt:
Atq + P1s
( r-5.s3)
and
Q2: A2q * P2s
( ls.s4)
Solution of Eqs. (15.-53)and (15.54)yields the valuesof q and s. Housel'smethod is
not widely used in rrractice.
Example15.7
The ultimate bearing capacity of a 700 mm diameter plate as determined from
field load testsis 280 kN/m2. Estimate the ultimate beaiing capacityof a circular
footingwith a diameterof 1.5m. The soil is sandy.
$
Solution
$
From Eq. (15.49),
$.
Blrooting)
:
Qugooting)
4rplut";
B(ptut*)
:r*(H)
: 680 kN/mz
Chapter 15 Soil-Bearing Capacity for Shallow Foundations
Example15.8
Following are the results of two plate load testsin a cohesivesoil:
Platesize
Settlement Totalload,O
(ft)
(in,)
(lb)
0.5
0.5
1 . 5x 1 . 5
2.5 x 2.5
15,750
33,750
If a squarefooting 5.75ft X 5.75ft is to be constructedand the allowablesettlement is 0.5in., what is the magnitudeof the total load that it can carry?
Solution
From Eq. (15.52),
Q=Aq*Ps
So
15,750= 1t.5)2++ (4 x 1.5),t
33,750= (2.5)2q+ (4 x 2.5)s
(a)
(b)
From Eqs.(a) and (b),
q : 3,0001b/ft2 and s : 1,500lb/ft
+ (4 x 5.75X1,500)
Q = Aq * ps : (5.75)2(3,000)
: 133.69kip
: 133,687.51b
15.9
Ultimate Bearing Capacity on Layered Soil
The ultimate and allowable bearing capacitiesol shallow foundations on weaker
(loose) sandsand soft clays can be increasedby placing a layer of compact (dense)
sand over it. This is essentiallya bearing capacity problem on a laycred soil, which
is the subject of discussionin this section.lt is divided into two parts - the first discussesthe bearing capacityon layered sand (denseover loose) followed by an evaluation of the bearing capacity on a stronger sand layer underlain by a weaker saturated clay layer.
Foundations on Layered Sand-Dense
over Loose
A simple theory for determining the ultimate bearing capacity of a foundation that
restson a layer of densesand underlain by loose sand has been proposed by Meyerhof and Hanna (1978).The basic principle of this theory can be explained with the
aid of Figure 15.18,which is for a strip foundation. When the top densesand layer is
15.9
l-<-
531
Ultimate Bearing Capacity on Layered Soil
zl -l
I
I
t
:::]:,.;,li
l,ii*ll
1lff;aJr;p:i$311ryt:Tr!q4:!1ffi;i:l$iis*'r'."*".-',1
Y'
Qr'
L
1f
Thinner
top layer
strong
sand
*'**-
gr:$i!i!rdr:{tg1iw:f
:'i
Strong
-"."...-l:iit]:.;l1'".,",..,", sand
Yr
0r'
c t ' =o
t
I
H
Thicker
top layer
Weaksrsqnd
'12'
sand
Weaker
"lz
Qz'
$.'
t2''Lo
ct'= 0
Figure 15.18 Bcaring capacityin layered sand- strong sand unclerlainby weak sand
relatively thick, as shown by the right-hand side of Figure 15.18,the failure surface
in soil under the foundation will be fullv located inside the densesand. For this case,
4 , , : 4 , t ) l t D t N , t ( t )+ ) l , B N r 1 1
( 1s.ss)
(for strip foundations)
4 , , : 4 , ( r ) : T t D t N q t t+ 0 ' 3 7 1 8 { 1 r 1
(for circular or square loundations)
( l.s.s6)
and
-'-(;)
4 , ,: 4 , , (-r 1v , D 1 l \ ' ' 1 , ;t I1 rt1 .
]r,nlurt't
(1s.s7)
where , -'::il:iffJffiil::'.il*sandinthiscase)
:
capacityfactors with referenceto the soil friction
Nqrrtand N"r,l
bearing
a n g l e ,r p i ( T a b l e s1 5 . 3a n d l - 5 . 4 )
. o w e v e r ,t h e
N o t e t h a t E q s . ( 1 5 . 5 5 ) .( 1 - 5 . 5 6 )a,n d ( 1 5 . 5 7 )a r e s i m i l a r t o E q . ( 1 5 . 2 8 ) H
be
somewhat
assumed
to
they
can
be
incorporated;
depth factors have not been
conservative.
If the thickness of the dense sand layer under the foundation H is relatively
thin, the failure in soil would take place by pLtnchingin the dense sand layer followed
532
Chapter 15 Soil-Bearing Capacity for Shallow Foundations
by a general shear failure in the bottom (or weaker) sand layer,as shown in the lefthand side of Figure 15.18.For such a case,the ultimate bearing capacityfor the foundation can be given as
Q u: e u @ ) *r , " ' (
./
2Dr\
t n
;
nn6'
-
)*,7
rrH < Qu(t)
t
IEq.(1-s.-5-s)]
(1,5.,51J)
(for strip foundations)
2 D ,\ / K . t a n d i \
^/
4tr-Qu(b1+2yfl'\1 *
u
H )[
/ol
ytH=q,,(,)
t
IEq.( 1- s.- s6) ]
( r,5.-se)
(for squarc or circular foundation)
and
/ . B \ . . . 1, 2 D i \ / K , t a nd i \ . .
r'n = Q,1,1
e u : Q , , t h (t +
l + i)t,u'1t * a /(
a
)^',-
t
(for rectangular
foundations)
lEq.(1s.s7)l
( 1-s.60)
where K. : punching sheetrcoeflicient
.,\i : shapefactor
eu1): ultimate bearing capacityo1 thc botttln-rsoil layer
can bc taken to be approximatcly l. Thc punching
The value of the shape factor ,tr'"
shear coefficientis
K,:
.f(yt,yz,l/rrrr,l/ri.t)
(1-5.61)
where 7, - unit weight of the lower layer of sand
N"Cl - bearing capacityfactor for the soil friction angle,rf!
. h e t e r m q , , 1 1 , ; iEnq s . ( 1 , 5 . 5 8 )(,1 - 5 . 5 9 ) ,
T h e v a r i a t i o n o f K i s s h o w n i n F i g u r e 1 5 . 1 9T
and (15.60)is given by the relationships
4 u ( b 1 : Y t ( D r+ H ) N q v z 1 +
)t.BNr1.1
(for strip foundations)
(1-5.62)
-r
4u@): Yr(Dt + H)N,te; 0'3Y28Nr,,1
(for circular or squarefoundations)
(15.63)
15.9 Ultimate Bearing Capacity on Layered Soil
k
o
)f)
€
E
=
E
0.4
0.(r
Y:Nyr2t
Figure 15.19
Variationof K. with (TzN'1z)l(y1Nr11)
T,Nr,,r
and
Q , , 1 t 1 :l ( D 1
+ r r ) N q ( 2')-. ,; ^f ( i ) 1 v , a x , 1 , 1
(for rectangularfoundations)
(1s.64)
Foundationson Denseor compacted sand overrying
soft ctay
If the thicknessof the sand layer unclerthe foundation
is relatively small, the
failure surfacemay extend intolhe soft clay layer.
This is shown in the left half of Figure 15'20' However, if the sand layer under tire fbundation
is large, the failure surface will lie entirely in the sand layer, as shown in
the right halt oFnigure 15.20.Ac_
cording to Meyerhof and Hanna (197g),in this case
the uttimate bearing capacityof
a strip foundation may be given by
Q,:
cuN,+ vT'z(t
.?)u,ry.
yDt
(1s.6s)
534
Chapter 15 Soil-Bearing Capacity for Shallow Foundations
clut
\u
$*o
Figure 15.20 Foundation on compacted sand layer ovcrlying sol't clay
with a maximum of
u,,: )tBlv, + YDll''1,
( 1.s.('6)
where d' : angle of friction of top sand layer
y : u n i t w e i g h to f s a n d
K. : punching shear resistancecocfficient
ly'" and N,, correspond to the anglc of friction, r/,', for sand (Tables l-5.3and l-5.4).
Noteifor d, : 0, N, - -5.14,as determined from Table 15.3.
For rectangularfoundations,
(r +i)rr'(t .?)*,'#
n,: (, +o.zf),,N"+
* rDr
(1s.67)
with a maximum of
q,:
r/
,\1
- oour)run, - yD,Nq
(15.68)
The variation of the punching shear resistancefactor, K., is given in Figure 15.21.
Equations (i5.66) and (15.68)are estimatesof the valuesof q,, for strip and rectangular foundations, respectively,in the upper sand layer. This condition corresponds
to that shown in the right half of Figure 15.20.
15.9 Ultimate Bearing Capacity on Lavered Soil
Figure 15.21
Variationof K, with ry''(according
to Meyerhof
a n dH a n n a )
0'(deg)
Example15.9
Figure 15.22showsa rectangularfoundationwith B : 4 ft and L = 6ft.using
a
factor of safetyof 3, determinethe net allowableload the foundation.un
"uriu.
F_r
=+ ft____r*l
layerof sandi,11"ry."
sinceit has@i: 42",whichisgrearerthan61 : 35..
Tl_"-t"O
Also,71 )
(15.60)
So Eq.
72.
should be used ro calculateq,:
Qu:Qu{b1*
(t * l)r,r,(t *+X5#9
^,- ,,,
536
Chapter 15 Soil-Bearing Capacity for Shallow Foundations
It is given that y1 : 118 lb/ft3, Tz : 105 lblf(, 6\:
T a b l e 1 5 . 4 , l y ' 7 1: r r 1 3 9 . 3 2a n d N r l z ; : 3 7 . 1 2 .S o
42o,and O'r: 35". Also, from
rzNylzl
(105)(37.12)
ffi
(,t-x"r-)
:
: 0'237
From Figure 15.19,for Qi : 4Z",thevalueof ff" is 6. Thus,from Eq. (15,60),
( 2 X 3 )l [ ( 6 ) ( t a 4
(
4\
...- - .t
n2")1,..
"
Qu:Qutb\+(1+6/(ttsXz.s),|t+;)L_'](l)-(l18X2.5)
: nu\b)+ 5349
(a)
From Eq. (1,5.64),
y,(4 + H)N,121.
r - o4(*)lr,ur,o,
eu(h):
*f
ZL
\L,/I
:33.3 and
*
From Tables15.3and 15. ,for Q2- 35o,the valuesof N,/12;
{12.1 37.15.
Hence,
Q , , ( , ,( :r 1 8 ) ( |3 2 . s X 3 3 . 3;)[ ' - r - ( : ) ] t r o r t t o l r r r . l s ): 2 t , 3 3 3 t b t r (
(b)
From Eqs. (a) and (b),
qu:27,333 + 5349= 32,682\blft2
(c)
We alsoneedto checkEq. (15.57):
* llt - o.o(l
eu: rtDrNo,,,
I L
\ L /
"1r,ary",,,
)
42",Nrtu : I39.32and Nqtrl: 85.38,so
From Tables15.3and 15.4,for 6\:
q , : ( t 1 8 X 3 X 8 s .(3+8))l t. - r . o ( l ) l t t t r r t o l3r te . 3 2 )
\z/L
\b,/t'
: 54,336lblft2
(O)
ComparingEqs. (c) and (d), q,,: 32,6821b/ft2,
we have
*
4
* rtDf :32,682 - (3X11S): ZZhZSbttt?
{,u{net1:
Q,
Ou :
Q,1n",1BL(32.328)(4)(6)
^
:
( ,*axr )
:258.6kips
r
15.9 Ultimate Bearing Capacity on Layered Soil
E xa mp l e1 5 .1 0
Referto Fisure15.20.For sand
y : 1I7 lb/ft3
Q' : 40"
and for clay
cu : 400lb lft2
For the foundation
.B = 3ft
L:4.5ft
4:3ft
H :4ft
Determinethe grossultimate bearingcapacityof the foundation.
Solution
The foundationis rectangular,so Eqs.(15.67)and (15.68)will apply.For g, : 4go,
from Table 15.4,N? : 93.69and
cuN,
(400xs.14)
:
: o'125
o^s?B^r,
0t(llDpxr3rr)
From Figure 15.2t,for c,N./0.SfBNn= 0.125and Q, :
K, * 2.5.Equation (15.67)gives
40o,the value of
q.:lrL +(0.2)(f)1.
+ (\ r +e)r.,,(
t .+)r,ry
'\L/)^
L/
"
r l l
/
r \
: || 1 + (0 .2/)(
+ { I + * lfr n) g) 2
*
)
I(
400Xs.t4)
L
4.5/.
\4.r/J
1
(2)r3)
|
lI/(, 2. s. ,), -u=n4 o + /t( 'ri"rl '7 x1 3 )
x l1 +
4
3
L1t-'"t
: 2330+ 5454+ 351: 8t35lb/ft2
;
I
Again,fromEq.(15.68),
1f
s": iLI * (t*)(f)]'u", * vDlNn
+yDr
Chapter 15 Soil-Bearing Capacity for Shallow Foundations
For 4 : 40",lfn = 64.20(Table 15.3)and
t- / r \l
q, - (0.s)l
+ (117X3X64.20)
1 (0.4)(
* ) ltrrz)t:lte:.6e)
\+.J,/ J
L
: 12,058+ 22,534= 34,5921b/ftz
Hence,
q, : 8135lb/ft2
15.10
Summary and GeneralComments
In this chapter,theories for estimatingthe ultimate and allowablebearing capacities
of shallowfoundationswere presented.Proceduresfor field load testsand estimation
of the allowablebearing capacityof granular soil basedon limited settlementcriteria
werc bricflydiscussed.
Severalbuilding codes now used in the United Statesand elsewhereprovide
presumptivebearing capacitiesfor various types of soil. It is extremely important to
realize that they are approximate volues only. The bearing capacity of foundations
dependson severalfactors:
1.
2.
3.
4.
5.
6.
7.
Subsoil stratification
Shear strength parametersof the subsoi.
Location of the ground water table
Environmental factors
Building size and weight
Depth of excavation
Type of structure.
Hence, it is important that the allowable bearing capacity at a given site be determined basedon the findings of soil exploration at that site, past experienceof foundation construction,and fundamentalsof the geotechnicalengineeringtheories for
bearing capacity.
The allowable bearing capacity relationships based on settlement considerations such as those given in Section 15.7do not take into account the settlement
causedby consolidation of the clay layers.Excessivesettlement usually causesthe
building to crack,which may ultimately lead to structuralfailure. Uniform settlement
of a structure does not produce cracking; on the other hand, differential settlement
may produce cracks and damage to a building
Problems
15.1 For the continuousfooting shownin Figure 15.23,determinethe grossallowable bearingcapacity.Use Terzaghi'sbearingcapacityfactorsand a factor of
safetyof 4. Assumegeneralbearingcapacityfailure.
Problems
Unit weight of soil = y
Dy
I
ll , fl
Df
L"#
I,'r'.,:.1.',*,t
F+-
Figure 15.23
B------------+l
Figure 15.24
a . y : 1 2 0 l b / f t 3 ,c ' : 0 , 6 , : 4 0 " ,D r : 3 f t ,
B : 3 . - 5f t
b . y : 1 1 5 l b / f C ,c , : 6 0 0 l b t f l c , 0 , : 2 5 . , D 1 :
3 . , 5f t , B : 4 f L
c. y:17.-5kN./mr.t,
: 2 0 . , D t : 1 . 0 m ,I l : 1 . 2 m
d . y : l 1 8 l b / f t . r , c: 4' -. 5_ 0l al tkrN
/ f{rm
r ,.d.:,62,g " , D r : 4 f t ,
B:4ft
e . y : 1 7 . 7k N / m r , c _ : 4 t 3k N / m 2 ,
e: g.,Dr:0.6m, B :0.ti m
15.2 Repeat Problem l5.l assuminglocal
shear failure.
t5'3 Redo Problem l-5.r using the Frandtr,
Reissner,and Meyerhot bearing capacity facrorsgiven in Tables l-5.3and l-5.4and
Eq. (l-5.2t3).'
15.4 A square footing has the following
values:
Cross allowahle load : 42.2601b
Factor of safety : 3
Dt:3tt
S o i l p r o p e r t i e s : y : 1 1 0l b / f t r
6 ' ,: 2 0 "
c' : 200lblft3
U s e E q . ( 1 - 5 . 1 2t)o d e t e r m i n et h e s i z eo f
the footing.
15.5 Repeat problem t5.4 for the following
data:
Gross allowable load : 1g70kN
Factor of safety : 3
D,: rm
y -- ll kNimr
6', :35"
c':0
15.6 A squarefootingis shown in Figure
15.24.Forthefollowing cases,determine
the gross allowable load,
Qn1',that the footing can carry. Use Terzaghi's
equation for general shear failure (F, : 3).
a . y : 1 0 5 1 b / f t 3 , 7 , u , :1 1 8 1 b / f t 3 ,:c0,, 0 , : 3 5 o ,
B:-5ft, Df :4ft,h:2ft
b. p : 1800kg/m3,p,o, : 19g0kg/m3,c, :
23.94tN1_r, Ol': 25",B : l.g m,
Dr:I.Zm,h:2m
Chapter 15 Soil-Bearing Capacity for Shallow Foundations
15.7 A square footing is shown in Figure 75.24.Use Eq. (15.28) for general shear
failure and a factor of safety of 3. Determine the safe gross allowable load.
Use the following values:
7 : 100 lb/ft3
7 , u t: 1 1 5 1 b / f t 3
c':0
Q' - 30"
B -- 4ft
Df : 3.5 ft
h:2ft
15.8 Solve Problem 1-5.7with the following values:
y : 15.72kN/ml
Y''.r: 18'-55kN/m3
c ' : 0
6', - 35'
B : 1.-5m
3
D1 : 1.22m
/r : 0.61 m
15.9 Solve Problem 15.7with thc foilowing data:
7 - 1 1 - 5l b i f t l
7"t : 122'4lblft3
c ' : l ( X )l b / f t 2
(t' : 30"
B - 4ft
Dt -3ft
h-4ft
15.10 The square footing shown in Figure l-5.25is subjectedto an eccentricload.
For the following cases.determine the grossallowable loerdthat the footing
could carry (use it, : j)'
r-------,
j
t
,
,
I
r
^
:+l
t
I
B
l
.
,
a -
a
Y l
_______L_:r___-
l<--B--------_--N
Figure 15.25
problems
541
a . y : 1 6 k N / m 3 , . .:' 0 , d , : 3 0 " ,B : 1 . 5 m ,D t : 1
m , r : 0 . 1 5 m , y: g
b . p : 2 0 0 0 k g / m 3r.l : 0 , 6 , : 4 2 o B
, : 2.5m,'Dr:1.5m,-r : 0.2m,y : g
c . p : 1 9 5 0 k g / m : ,c ' : 0 ,
e, : 40.,8: 3 m,Dt:1.+^,i:
0 . 3m , y : 6
15.11 For a square footing supported by a sand,given
that B : 2 m, Df : 7.5 m,
corrected standardpenetration number N., :
9, allowable settlementS" :
20 mm, estimate the net allowable bearingcapacity.
15'12 A platc load test was conducted in a san<1y
soil in which the size of the bearing plate was 1 ft x l ft. The ultimate load per
unit area (q,,) forthe test was
found to be 4200lblft2. Estimate the total altowabt.
toua ([.,,; tbr a footing
of size -5.5ft X -5.-5
ft. Use a factor of safetyof 4.
15.13 A plate load test (bearing prate of 162mmdiameter)
was conductecrin clay.
The ultimate road per unit area, q,,,for the test
was found to be 200 kN/m2.
w h a t s h o u l d b e t h e t o t a r a l r o w a b l er o a c r ,
e . , 1 1f o, r a c o l u m n f b o t i n g 1 . 7 - m
5 in
diameter'/ Use a factor of safetvof 3.
15'14 The resultsof two fieltl loacltc-stsin a clay
soil are given in the tollowins table:
Plate diameter
{mm)
Settlement
(mm}
204.8
151.2
l-s
Total load
(kN)
49.-5
133.1
I-5
Based on thcse rcsurts,clcterminethe size of a
squarc footing that wilr carry
a t o t a l l . a d o r ' 3 0 0k N w i t h a m a x i m u m s c t t r e m e n t
of 15mm.
15'15 Figurc l-5.26showsa footing on layered
sancl.Determine the net allowable
l o a d i 1c a n c a r r y ,g i v e nt h e f b l l o w i n ec o n d i t i o n s :
S q u r r r cl i r o t i n g :6 : . 5l i
Factor o1 safcty requircd : 4
D/ - 3.-5ft
lI:2It
- | lXlhlfr'
ft
f : - 1 0 5t b i t i r
<b\ : 40"
6\ - 30"
I
I
J
Sand
i 1 :: :
'
Tr
Qr'
''
saiid"l
'Vt
Qz,,
Figure 15.26
Chapter 15 Soil-Bearing Capacity for Shallow Foundations
15.16 Redo Problem 15.15with the following values:
Rectangularfooting: B : 1 m; L : 1.5m
Factor of safetvreouired : 3
D1:7m
H :0.6m
7r - l7'5 kN/ml
y z : 1 . 5k N / m l
0\: 42
oi: 34'
1 5 . 1 7R e f e r t o F i g u r e1 5 . 2 0 . T h e f o u n d a t i o n1i sm x 2 m i n p l a n . D 1 : l m a n d
H : 1 . 5m . F o r t h e s a n d l a y e r ,@ ' : 3 5 o ,c ' : 0 , y : 1 7 . 8k N / m r ; a n d , f o r t h e
clay layer, d : 0'. c,, - 60 kN/m2, y : 18.2kN/m3. Detcrmine the grossallowable load that the foundation could carry. Use F, - 4.
1 5 . 1 8 R e d o P r o b l e m 1 5 . 1 7w i t h t h e f o l l o w i n gd a t a :
x 6ft
F o u n d a t i o n :B x L : 3 f t
Dt : 2.5 tt
H:3tt
Sand: ,h' : 40'
(:' : U
v:
Clay:
q)
Ll,,
v
I 15lb/frr
-0'
: 7-501b/ft2
- 1 1 Ul b / f t 1
References
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il
I