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BEARING CAPACITY OF ECCENTRICALLY OBLIQUELY LOADED FOOTING
By Swami Saran1 and R. K. Agarwal2
ABSTRACT: The bearing capacity of an eccentrically obliquely
loaded footing is determined by limit equilibrium analysis. The
footing is considered rigid with a rough base. It is assumed that
the rupture surface is a log spiral and that failure occurs on the
same side as the eccentricity, with respect to the center of the
footing. The resistance mobilized on this side is fully passive and
partial on the other. The footing is assumed to lose contact with
an increase in eccentricity; the results are given in the form of
bearing capacity factors, JV7, 7V(/, and TV,.. For the verification
of analytical solutions, model tests were conducted on sand.
Footings were tested both at the surface and at a depth such that
DfIB = 0.5, eccentricity of load ranged from Q.IB to 0.35, and
inclination of load varied from 5 to 20, in which Df and B are,
respectively, depth and width of footing. The results of the
previous inves-tigators are also analyzed and compared with the
proposed theory. A reasonable agreement was found between the
theory and the test data.
INTRODUCTION
Footings serving as foundations for retaining walls, abutments,
stanchions, and portal-framed buildings may be subjected to moments
and shears in addition to the vertical load. These forces and
moments may be replaced by an eccentric-inclined load on the
footing. The usual practice to design such footings is to resolve
the eccentric-inclined load in two parts, namely: (1) An eccentric
vertical load; and (2) a central oblique load. The bearing capacity
of the footing is then obtained by analyzing the problem in two
separate parts: (1) The bearing capacity of footing subjected to
eccentric vertical load; and (2) the bearing capacity of footing
subjected to central oblique load. The two values of bearing
capacities thus obtained are su-perimposed to get the bearing
capacity of footing subjected to eccentric-inclined load.
Many investigators have studied the problem of footing subjected
to ec-centric vertical load (Meyerhof 1953; Eastwood 1955; Jumikis
1961; Dhillon 1961; Zaharescu 1961; Prakash and Saran 1971;
Purkayastha and Char 1977; Purkayastha 1979). Footings subjected to
central oblique loads are also analyzed by many investigators
(Schultze 1952; Meyerhof 1953; Hansen 1955; Kezdi 1961; Hansen
1961; Muhs and Weiss 1969; Saran 1971; Saran et al. 1971; Kameshwar
Rao and Murthy 1972; Muhs and Weiss 1973; Hanna and Meyerhof 1981).
A critical review of all these methods is available elsewhere
(Agarwal 1986). There seems no investigator who analyzed the
problem of a footing subjected to eccentric inclined load. However,
a few investigators have studied this with the help of model tests
(Meyerhof 1953; Saran and Niyogi 1970).
The purpose of this investigation is to develop a rational
theory for the computation of the ultimate bearing capacity of
eccentrically obliquely loaded
'Prof, in Civ. Engrg., Univ. of Roorkee, Roorkee-247 667, U.P.
India. 2Reader in Civ. Engrg., Inst, of Tech., B.H.U., Varanasi,
India. Note. Discussion open until April 1, 1992. To extend the
closing date one month,
a written request must be filed with the ASCE Manager of
Journals. The manuscript for this paper was submitted for review
and possible publication on September 15, 1990. This paper is part
of the Journal of Geotechnical Engineering, Vol. 117, No. 11,
November, 1991. ASCE, ISSN 0733-9410/91/0011-1669/$l.00 + $.15 per
page. Paper No. 26316.
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footings. The problem is solved by two methods: (1) Limit
equilibrium analysis; and (2) limit analysis. In this paper the
analysis done by limit equilibrium method is presented. In this
study, model tests were also in-cluded to evaluate the proposed
theory. In addition, test data of the previous investigators were
also analyzed, and compared with the proposed approach.
This is the first attempt to develop a rational theory for
predicting bearing capacity of eccentrically obliquely loaded
footings.
T H E O R Y
The following assumptions have been made in the analysis. [Note:
In Appendix II the dimensions of symbols are given in F (force), L
(length), and T (time).]
First, a shallow strip footing with a rough base is considered.
The weight of the soil above the base of the foundation is replaced
by uniform surcharge.
Second, one-sided failure is assumed to occur along the surface
AEDC (Fig. 1). The failure region is divided into three zones I,
II, and III. Zone I is an elastic zone. Zone III is passive Rankine
zone. Zone II is located between zones I and III and is known as
zone of radial shear. The curved portion ED is a log-spiral having
its center at the edge of the footing (B) (Saran 1970).
Logarithmic spiral is represented by the following equation ,- =
,-()eeitan,|, ^
where rQ = initial radius of log-spiral equal to BE; r = radius
of log-spiral at an angle 0, measured from rQ (i.e., BE) in the
clockwise direction; and
HB
-
4> = angle of internal friction. The sides of the elastic
wedge are inclined at angles a, and a2 with the horizontal.
Third, a similar rupture surface is considered when the footing
loses contact with the soil due to excessive eccentricity. The
rupture surface starts from a point A' instead of A [Fig. 1(b)].
The effective width is represented by Bxy, where x, is the ratio of
contact width to total width of the footing.
Fourth, the soil on the right side of the failure plane AE or
A'E [Fig. l(a and b)] gets partially mobilized, and this is
characterized by a mobili-zation factor m. Shear strength of the
soil is then expressed as T = mc + u tan 4>, (2a) where 4>, =
tan ~ l(m tan ) (2b) To compute partial resistance offered by this
side a rupture surface shown by dotted lines is considered.
Fifth, superposition of limit stresses for three cases: (1) c =
q = 0; (2) c = y = 0; and (3) q = y = 0 holds; in which c =
cohesion; -y = density of soil; and q = the intensity of surcharge
(Fig. 1).
Analytical solutions are developed for a general case in which
the footing has lost some contact with the soil. The contact width
of footing A'B is assumed to be Bxl [Fig. 1(b)]. For full footing
contact, xx = unity. Solutions are developed as follows.
Geometry of Failure Surface The various sides of the failure
surface are expressed in terms of footing
width B, angle of internal friction 4>, wedge angles a, and
a2, and contact width factor xx by considering the sine rules for
triangle A1 BE and properties of log spiral.
The various sides can then be expressed in terms of
nondimensional terms 0!, x, and y, defined as
sin a2 , . x = , : (3)
sin(a! + a2) y = -iE^-) W
sin(a, + a2) BE = Bxxx = ra (5) A'E = Bxxy (6) BD = r =
r0ee>'""* = Bxjjce0"11"4, (7)
HD = BD sin! 45 ~ j) = 5x,xee""n sinU5 - ~) (8)
BH = BD cos( 45 - | J = fix^e0"""* cos( 45 - ^ J (9)
Bearing Capacity Expression The bearing capacity expression is
then developed by considering the
equilibrium of elastic wedge A 'BE. The forces acting on the
wedge include
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earth pressures P, and Pp on sides A 'E and BE; adhesion force
Ca and C'a on BE and A'E; and eccentric inclined load Qd (Fig.
2).
Neglecting the weight of soil wedge A'BE, footing equilibrium
requires that Qrf cos i = Pp cos(a! - ) + P, cos(a2 - ,) + Ca sin
aj + C'a sin a2 (10) If c = unit cohesion, then C = c5ii; and C/, =
mcA'E. Substituting the values of BE and A'E from (5) and (6) into
the preceding expressions
sin(a! + a2) v ;
c , = mc^xt sin a t sin(ai + a2) ^ '
are obtained. Substituting the values of C and C^ from (11) and
(12), respectively,
into (10) Qrf cos / = Pp cos(a, -
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Pmc, representing the forces due to weight, surcharge, and
cohesion, re-spectively, of the soil mass considered.
Thus the value of the bearing capacity may be calculated by
replacing Pp and P, in (13) by Ppy + Ppq + Ppc and Pmy + Pmq + Pmc,
respectively. Thus
Qd cos i = (Ppy + Ppq + Ppc)cos(a! - c|>) )cos(a2 - ,)
(1 + m)cBxx sin a, sin a2 , s + r-^ 7 (14)
sin(ai + a2) Surcharge intensity q can be expressed as
q = yDf (15) in which Df = the depth of foundation (Fig. 1). By
introducing
2[Pp7 cos(ax -()>) + P,, cos(a2 - m)] W7 - 772 : ( I D )
yBz cos i Nq = -& \[n _"] x z ^ (17) Ppq cos ( a i - ) + Pmq
cos(a2 - ()),)
yDf cos i
Ppc cos(a! - ) + P,c. cos(a2 - ,) yDf cos i
P.._ cosCa, - < Nr = cB cos i
(1 + m)x, sin a, sin a2 n ^ + - . . ^ (18)
sin(a! + a2)cos ; into (14)
Qrf = BU yBNy + yDfNq + cNc) (19)
is obtained, in which quantities Ny, Nq, and Nc are called the
bearing capacity factors. These quantities are dimensionless and
depend on 4>, e/B, and (' only.
Computation of Earth Pressure Ppy, Ppq, and Ppc Passive earth
pressures Ppy, Ppq, and Pp c are determined by considering
the equilibrium of the soil mass BEDH. The forces, which are
considered in the determination of passive earth pressure, on the
wedge BEDH are enumerated in the following and shown in Fig. 3.
1. Passive earth pressure Py due to weight of soil CHD (Fig. 1)
acts hori-zontally at height HD/3 from the point D.
2. Weight W of the soil mass BEDH acts vertically downward at
the center of gravity of soil mass BEDH.
3. Passive earth pressure Pq due to surcharge on CH (Fig. 1)
acts horizontally at a height HD/2 from the point D.
4. Surcharge weight XVq acting on BH: The intensity of surcharge
q = yDf, where 7 is the density of soil and Df the depth of
footing, acts uniformly dis-tributed over BH.
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FIG. 3. Forces on Soil Mass BEDH
5. Passive pressure Pc due to cohesion on soil mass CHD (Fig. 1)
acts horizontally at a height HDI2 from the point D.
6. Cohesive force C acting along arc DE. 7. Cohesive force C
acting along the face BE. 8. Passive earth pressure Ppy acting on
face BE acts at a lower third point
of BE at an angle c|> anticlockwise with normal at that
point. 9. Passive earth pressure Ppq acting on face BE acts at the
midpoint of BE
at an angle anticlockwise with normal at this point. 10. Passive
earth pressure Ppc acting on face BE acts at the midpoint of BE
at an angle anticlockwise with normal at that point. 11.
Resultant F of normal and frictional forces passes through the
center
of the log-spiral, since it makes an angle with the normal at
the point of application.
The passive earth pressures, Ppy, Ppq, and Ppc, are determined
by taking the moments of all the forces about the center of the
log-spiral (i.e., at the edge of footing). The moment of the force
F gets eliminated, since it passes through the center of the log
spiral.
Moments of all the forces taken about the center of the
log-spiral are given in (20)-(28), as shown in Table 1.
For equilibrium of soil mass BEDH, SM = 0; or
(M l 7 + M2y) + (Mlq + M2ll) + (Mlc + M2c) = PpyBT, + PpqBT2 +
PpcBT2 (29) or
NpyyBi + NpqqB2 + NpccB2 = P^BT, + PpqBT2 + PpcBT2 (30)
where
= Mly + M2y V Y
yg3 V1)
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TABLE 1. Force and Moment Equations Serial
number (D
1
2
3
4
5
6
7 8 9
10 11
Force (2)
P-,
W
P.,
% P,
c
Ci p
1 l>y P ' l"l
P,n F
Moment of force listed in column 2 about center of log spiral
(Fig. 3)
(3)
Mh = 7B3 - .vV"* cos
-
Ppc = - cB (366)
The solution of P , P , and Ppc thus obtained was used in
solving bearing capacity factors defined in (16), (17), and
(18).
Computation of Earth Pressures Pmy, P,m/, and PIC The values of
passive earth pressures Pmy, Pup and PIC at a mobilization
factor m can be obtained by substituting the angle of internal
friction by 4>, [obtained from (2b)] and changing the wedge
angle a t to a2 and a2 to ! in (20)-(36).
Relationship between Wedge Angles ax and a2 The relationship
between wedge angles a! and a2 is obtained by solving
the three equilibrium equations obtained in three cases
separately (Fig. 4). The equations so obtained are for soils having
weight only (c = q = 0),
Fig. 4(a): Eliminating Ppy, P,7, and Qdy from three equilibrium
equations, we get
(c) q = y = 0 FIG. 4. Equilibrium of Elastic Wedge A'BE
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2 sin(oii A i) 3 sin(a, + a2 A 4>m)
sin(a2 A, + ;') sin(tt! + a2 - 4> - A,)
sin(ax + a2)
cos A sin a2 3 sin a :
cos(a, + a2 A)
cos i 1 + 1
sin a , \ Bxx 2xx) After simplifying (37) we get a quadratic
equation as follows: Ay tan2a2 + By tan a2 + Cy = 0
or
(37)
(38)
tan a2 By V{By)z - 4AyCy
2A~
where
Ay = cos A cos(A, /) 2 sin a{ cos(Am - /)sin(a! - A)
e 1 - 3 cos i cos a! c o s ^ A A,)l 1 +
\ IJX i 2.X i J
+ 2 sin a! cos a, sin(a, A j) By = 3 sin ! cos(a! - A - a, + i)
- cos A sin(A, i)
1 3 cos l sin(2a] A Am) 1 +
\ Bxi 2xx Cy = 2sin(ax ()> /)cos A^sincq Ssino^si^A,,,
i)cos(a! A)
e 1 3cosi 'sina, sin(a, A Am)l 1 + 1
Bxx 2xx
(39)
(40)
(41)
(42)
For soil having surcharge only (7 = c = 0), Fig. 4(b):
Eliminating Ppq, Pmq, and Qtlq from the three equilibrium
equations, we get
sin(a! A i)cos Am sin(a2 Am + /') cos A sin a2 2sin(a t + a2 A -
Am) ' sin(ax + a2 A - A,) 2 s ina !
.sin(a! + a2) . e 1 + cos(a! + a2 - A) = c o s 1 1 Bxx 2xx s i n
a j
Simplifying (43a), we get a quadratic equation, as follows: Aq
tan2a2 + Bq tan a2 + Cq = 0
or
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(43a)
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tan a , = Bq \JB\ - 4AqCq
2A
where
Av = cos c|> cos(c()I - /) - 2 sin ctj sin(a, -
c|))cos((j>, - i)
1 2 cos i[ 1 + cos a, c o s ^ cf>m)
Bxl 2xJ + sin (X[ cos 4>, sin(a, - 4> /) Z?9 = 2 sin a,
cos(aj - c|> - ((>, + /) - sin(cj), - (')cos
2 cos / sin(2a, 4> ~ 4 0 1 1 + #*! 2x,
C9 = sin a, cos 4>, sin(a, , i')cos(a1 cj>)
2 cos z sin a, sin(ax c|> 4>,)l 1 At , 2* For the soils
having cohesion only ((7 = 7 = 0)
N,c cos (\> N, cos cp sin a2 2 sin o^
Nc cos 2 . sin(a2 + a2)
sin a.
+ cos(a! + a2 )
1
(44)
(45)
(46)
(47)
+ x, sin a?
fix, 2x1 (48)
Eq (48) is solved by trial and error. The detailed derivations
of the foregoing expressions are given elsewhere
(Agarwal 1986).
1.2
0.8
0-4
0
( a ) (b) IN
= i / 6 :
v
\ \
\
( c )
0 0-2 0-4 0.0 0.2 0.4 0.0 0-2 0.4 0.6 e/B e/B e/B
FIG. 5. Variation of x, with elK
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COMPUTATION
The range and interval of variables employed in computing
bearing ca-pacity factors are given in Table 2.
The following steps indicate the manner in which computations
are made for a given value of angle of internal friction, e/6,.and
/.
1. The value of xt is determined from the assumed contact width
variation, as given in Fig. 5.
2. A particular value of mobilization factor in is assumed. 3.
The values of wedge angles a2 for assumed values of wedge angle a,
are
determined from the wedge angle relationships previously
developed. 4. For one set of wedge angles a, and a,, the values of
passive earth pressures
PP7/(7S2) and P/(7S2); YJ{qB) and P , ,%5) ; and Pp(7(cS) and
PJ(cB) were determined for three different cases.
5. The values of the passive pressures thus computed satisfy the
two conditions 2V = 0 and 1M = 0 simultaneously, because the former
is used for determining bearing capacity Qd and the latter is used
in developing the wedge angle rela-tionship. Only the equilibrium
equation, 1H = 0, remains to be satisfied. The values of the earth
pressures computed in step 4 are substituted in the equilibrium
equation, 1H = 0. If the equilibrium equation ( 2 / / = 0) is
satisfied, the values of a, and a2 adopted in the computations are
in order.
6. Steps 3-5 are repeated for different values of mobilization
factor m. The passive pressures for maximum value of m satisfying
the equilibrium conditions, tH = 0; IV = 0; and 2M = 0 are
adopted.
The maximum value of m is chosen because, for failure, the soil
must develop maximum possible resistance compatible with stability.
The cor-responding bearing capacity factors are smallest in this
case.
MODEL TESTS
Model tests were performed on dry Ranipur sand (Dl0 = 0.13 mm; C
= 2.10; and Gs = 2.66) at relative density (RD) of 84%. Angle of
shear resistance was determined from slow triaxial tests to be
41.
The tests were conducted in a tank (150 cm x 150 cm x 100 cm
deep) on footings of sizes 20 cm x 20 cm, 20 cm x 40 cm, and 10 cm
x 60 cm. On each footing 20 tests were performed keeping the base
of the footing on the top surface of sand (i.e., DfIB = 0) for
different values of inclination of load (0, 5, 10, 15, and 20) and
amount of eccentricity (e/B = 0, 0 .1,
TABLE 2. Range and Interval of Variables Used In Computation
Variable (D
-
e/B i
m
x, Three as
Range (2)
10-40 0-0.3
0-30 0-1.0
types of variations shown in Fig. 5.
Interval (3) 5 0.1
10 0.2
Three types of variations as shown in Fig. 5.
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Roller
-Foot ing wi th ver t i ca l horizontal displacement measur ing
device and t i l t meter at top
Tank 1500x1000 x 1000
Foundation f o r r e a c t i o n l oad ing
( All d i m e n s i o n s in mm )
FIG. 6. Details of Loading Arrangements
0.2, and 0.3). An additional 20 tests were conducted on a
footing 10 cm x 60 cm, placing its base 5 cm below the top surface
of the sand (i.e., DfIB = 0.5).
The load on the footing was applied by means of a screw jack
caliberated through a proving ring. Arrangements were made to
observe accurately the settlement of the point of load application,
lateral displacement, and tilt of footing for each increment of
load. The complete setup is shown in Fig. 6.
INTERPRETATION
Of five assumptions made in the development of the analysis,
three need evaluation, viz assumptions listed at serial 2,3, and 4.
The other assumptions are commonly used in all bearing capacity
computations.
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ia
40
30
20
10
V ' / x/* '
I AY Iff ^ ^ j ^ i
i = 10 (b ) '
.
40
30 40
FIG. 7. Nr versus <
The pressure at the base of the eccentric-inclined loaded
footing are higher in magnitude on the side on which the load acts.
The settlements are there-fore likely to be larger on this side. In
this manner, with increase in load the soil on this side passes
into the plastic state first and the footing is likely to fail by
tilting. In all the tests conducted in this study it was observed
that the rupture surface develops on one side only in the direction
of horizontal component of the inclined load. Similar observations
were made by previous investigators (Meyerhof 1953; Eastwood 1955;
Dhillon 1961; Zaharescu 1961; Lee 1965; Saran et al. 1971). All the
observations justify the as-sumption of one-sided failure.
It has been established that the failure occurs on one side
only. However, some pressures do develop on the other side as well.
At equilibrium the resistance developed on the other side will not
reach the full mobilization value. Hence the pressure on this side
is considered at partial mobiliza-tion of strength for computation
of bearing capacity.
According to the second assumption in the section headed
"Theory," the center of the log-spiral is taken at the edge of the
footing. Saran (1970) showed that for the footing subjected to
central vertical load the log-spiral is tangential to the vertical
only when the center of log-spiral lies on the line EB or its
extension. On varying the center of log-spiral on line EB or its
extension, it is found that minimum values of the bearing capacity
factors Ny, Nq, and 7VC come when the center of log-spiral
coincides with the edge
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40
30
20
10
0
1 / /ats
III
1 ,
'" ^^" '
-
i = 2 0
( c )
40
30
20
10
0
1///
i
i = 30
i
1
"
W .
20 Nq
10 20 Nq
FIG. 8. Ntl versus
of footing. Further, it gives only one rupture surface for all
the three factors. Because of this, the center of the log-spiral
was kept at the edge of the footing.
According to the third assumption, the footing loses contact due
to ex-cessive eccentricity. The effective width of the footing is
represented by Bx{, where xx is the ratio of contact width to the
total width of the footing. This assumption is quite logical.
However, a variation of x{ should be con-sidered in the proper way.
In the present analysis three types of contact width variation are
considered, viz triangular variation, conventional vari-ation, and
full-width variation (Fig. 5). It has been observed that the values
of bearing capacity factors Ny, Nq, and Nc do not depend on the
contact width variation (Agarwal 1986). However, the length and
shape of the rupture surface is changed. This was also observed by
Saran (1969).
Bearing Capacity Factors The bearing capacity factors Ny, Nq,
and Nc are presented in Figs. 7-9.
It is evident from these figures that all three factors follow
the same trend with respect to cj), e/B, and i. These factors
increase with an increase in 4>, and decrease with an increase
in e/B and /.
Comparison with Model Test Data Fig. 10(a) and Fig. 10(b) show
the comparison between the ultimate
bearing capacity obtained from the proposed theory and model
tests, re-spectively, for DfIB = 0 and 0.5. These figures indicated
that the experi-
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4 0
30
20
10
0
- //y& / / ASA / / / A If
III i i
i*= 0
,
( a )
20 40 60 BO 100
CD O
K)
40
30
20
10
0
X / / : / y y / / /%S^
1 //\r
ill i = 10
( b )
10 20 30 40 50 60
en a X)
!S
4U
30
20
10
0
/ AA&^ - / /Ax
/ / A'v lllf i l l
1 1 1 I r
= 2 0
( c )
i
10 20 30
Dl dl
is
4U
30
20
10
0
////
If III i = 30
1
( d )
10 20 30
Nc
FIG. 9. Nr versus 4>
ca
paci
ty
O
I 2 0
S IO D
e
^ 0
/
/
. ; r ---^ r (*) -/
2.0 3.0 Exper imenta l b e a r i n g capac i t y { k g/cm )
FIG. 10. Experimental and Computed Values of Bearing
Capacity
mentally obtained ultimate bearing capacity compares well with
the ultimate bearing capacity obtained by the proposed theory.
Comparison of Bearing Capacity Factors Table 3 compares the N7
values obtained from the proposed analysis and
those obtained from other solutions for the footing subjected to
central vertical loads (i.e., elB = 0.0; i = 0).
The Ny values obtained for footings subjected to a central
vertical load (e/B = 0.0; z' = 0.0) are larger than the value of
Terzaghi (1943), Meyerhof 1683
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TABLE 3. Comparison of Ny Values for Footing under Central
Vertical Load (elB = 0.0;; = 0.0)
Degrees (D 20 30 40
Present analysis
(2) 6.36
29.45 166.15
Terzaghi (1943)
(3) 5.00
19.70 100.40
Ny Values
Meyerhof (1951)
(4) 2.80
16.00 95.00
Sokolovski (1965)
(5) 3.16
15.3 85.3
Prakash and Saran
(1971) (6) 3.82
19.42 115.80
Saran (1971)
(7) 6.05
29.35 165.26
(1951), Sokolovski (1965), and Prakash and Saran (1971). They
are almost equal to Saran's (1971) Ny values.
It is generally known that Terzaghi's (1943) values give
conservative es-timates. Experiments performed on models and at
full scale by Muhs and Kahl (1954), Feda (1961), Selig and Mckee
(1961), and De-Beer (1965) showed that Terzaghi's analysis
underestimates the bearing capacity. Saran (1969,1971) showed by
analyzing model test data that the values of Terzaghi, Meyerhof,
and Sokolovski give lower-bound estimates of Ny values. The
difference is significant for higher values of (((> > 20).
Hence, the Ny values in the proposed analysis may be more nearly
realistic.
Fig. 11 compares the Ny factors obtained from the present
approach with those of previous investigators. Fig. 11 indicates
that the proposed Ny values compare reasonably with Meyerhof
(1956), Prakash and Saran (1971), and Saran (1971). Hansen's (1961)
empirical relation underestimates the Ny values. Muhs and Weiss's
(1973) empirical reduction factor, for = 40, gives higher values
except for cj> = 30, where the values are very close to the
proposed value. Vesic's (1970) relation indicates lower Ny values
than the proposed Ny values.
Table 4 compares the Nq values obtained from the proposed
analysis with those of Terzaghi, Meyerhof, Sokolovski, and Prakash
and Saran for foot-ings subjected to a central vertical load (e/B =
0.0; i = 0.0).
Values of Nq obtained from the present analysis are the same as
those obtained by Terzaghi (1943), Prakash and Saran (1971), and
Saran (1971), but are higher than Meyerhofs (1951) and
Sokoloviski's (1965). This is because the rapture surfaces assumed
by Meyerhof and Sokolovski pertain to smooth footings.
Fig. 12 compares the Nq values obtained by previous
investigators with those of the proposed theory.
It is clear from Fig. 12 that Nq values obtained using Meyerhofs
reduction factor give higher values for = 40, but Nq becomes very
close to the proposed value of = 30. Hansen (1961) and Vesic (1970)
show the same trend because it was obtained using Meyerhofs (1951)
relation. Saran's (1971) factor for central inclined load for = 40
is higher than the proposed value but very close to the proposed
value for = 30. Prakash and Saran's (1971) factors for eccentric
vertical load compare very well.
Table 5 compares the Nc values obtained from the proposed
analysis with those of Terzaghi, Meyerhof, Sokolovski, Prakash and
Saran, and Saran for footings subjected to a central vertical load
(elB = 0.0; i = 0.0).
Values of Nc obtained from the present analysis are almost the
same as those obtained by Terzaghi (1943), Prakash and Saran
(1971), and Saran
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** 4.0 a
3.0
a a o
(J = 30" Meyerhof ( 1956) Hansen ( 1961 ) A Vesic (1970) X
Prakosh 4Saran(l971) S a r a n ( 1971) 4 Muh
-
FIG. 12. ^-Factor (Proposed Method) and Ntl Factor (Other
Investigators)
TABLE 5. Comparison of Nt. Values for Footing under Central
Vertical Load (e/B = 0.0; / = 0.0)
Degrees (1) 20 30 40
Present analysis
(2) 17.71 37.24 95.98
Terzaghi (1943)
(3) 17.70 37.20 95.70
Nc Values
Meyerhof (1951)
(4) 14.50 31.00 73.00
Sokolovski (1965)
(5) 14.80 30.1 75.3
Prakash and Saran
(1971) (6)
17.30 36.60 94.83
Saran (1971)
(7) 17.50 37.20 95.40
theory. Saran's (1971) Nc values for central oblique load are
higher for = 40, but are very close to the proposed Nc values for =
30.
CONCLUSIONS
A theory was proposed for determining the bearing capacity of
eccen-
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0 20 40 60 80 B e a r i n g c a p a c i t y f a c t o r N c ( p
r o p o s e d t h e o r y )
FIG. 13. /V,-Factor (Proposed Method) and Nc Factor (Other
Investigators)
trically obliquely loaded strip footings having a rough base
using the concept of one-sided failure. The results are given in
terms of nondimensional bear-ing capacity factors that depend only
on (j>, e/B, and /.
Comparison of the proposed theory with the model test data
showed excellent agreement. The results of the proposed theory were
compared with the work of previous investigators and found in good
agreement.
APPENDIX I. REFERENCES
Agarwal, R. K. (1986). "Behavior of shallow foundations
subjected to eccentric-inclined loads," thesis, presented to the
University of Roorkee, at Roorkee, India, in partial fulfillment of
the requirements for the degree of Doctor of Philosophy.
Chen, W. F. (1975). Limit analysis and soil plasticity. Elsevier
Scientific Publishing Co., London, England.
De-Beer, E. E. (1965). "Bearing capacity and settlement of
shallow foundations on sand." Proc, Symp. Bearing Capacity and
Settlement of Foundations, Duke Univ., Durham, N.C., 15-34.
Dhillon, G. S. (1961). "The settlement, tilt and bearing
capacity of footings on sand under central and eccentric loads." /.
Nat. Building Organisation, New Delhi, India, 6, 66.
1687
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Drucker, D. C , and Prager, W. (1952). "Soil mechanics and
plastic analysis or limit design." Q. Appl. Maths., 10(2),
157-165.
Eastwood, W. (1955). "The bearing capacity of eccentrically
loaded foundation on sandy soils." The Structural Engineers,
London, England, 33(6), 181-187.
Feda, J. (1961). "Research on bearing capacity of loose soil."
Proc, 5th Int. Conf. on Soil Mech. and Found. Engrg., Paris,
France, Vol. I, 635-642.
Hanna, A. M., and Meyerhof, G. G. (1981). "Experimental
evaluation of bearing capacity of footing subjected to inclined
loads." Canadian Geotech. J., 18, 599-603.
Hansen, J. B. (1955). "Single Beregning of Fundamentern
Baereevne" (in German). Springer-Verlag, Berlin, Germany, 229.
Hansen, J. B. (1961). "A general formula for bearing capacity."
Bulletin No. 11, Danish Geotechnical Institute, Copenhagen,
Denmark.
Jumikis, A. R. (1961). "Shape of rupture surface in dry sand."
Proc, Int. Conf. Soil Mech. and Found. Engrg., Dunod, Paris,
France, 1, 693.
Kameshwara Rao, N. S. V., andKrishnamurthy, S. (1972). "Bearing
capacity factors for inclined loads." J. Geotech. Engrg. Div.,
ASCE, 98(12), 1286-1290.
Karal, K. (1977). "Application of energy method." J. Geotech.
Engrg. Div., ASCE, 103(5), 381-397.
Kezdi, A. (1961). "The effect of inclined loads on stability of
foundations." Proc, Vth Int. Conf. on Soil Mech. and Found. Engrg.,
Dunod, Paris, France.
Lee, I. K. (1965). "Footings subjected to moments." Proc, VI
Int. Conf. Soil Mech. and Found. Engrg., University of Toronto
Press, Toronto, Canada, 2, 108.
Meyerhof, G. G. (1951). "The ultimate bearing capacity of
foundations." Geotech-nique, London, England, 2, 301-332.
Meyerhof, G. G. (1953). "The bearing capacity of foundations
under eccentric-inclined loads." Proc, 3rd Int. Conf. on Soil Mech.
and Found. Engrg., Zurich, Switzerland, 1, 440-445.
Meyerhof, G. G. (1956). "Penetration tests and bearing capacity
of cohesionless soils." J. Soil Mech. and Found. Div., ASCE,
82(1).
Muhs, H., and Kahl, H. (1954). "Ergebnisse V on Probebelastungen
and grossen Lastflachen Zur Ermittlung der Bruch last in Sand."
Mitteilungen der DEGEBO (in German), (8).
Muhs, H., and Weiss, K. (1969). "The influence of load
inclination on bearing capacity of shallow footings." Proc, 7th
Int. Conf. on Soil Mech. and Found. Engrg., 187.
Muhs, H., and Weiss, K. (1973). "Inclined load tests on shallow
strip footings." Proc, 8th Conf. on Soil Mech. and Found. Engrg.,
Moscow, U.S.S.R., Vol. 1, Part III, 173.
Prakash, S., and Saran, S. (1971). "Bearing capacity of
eccentrically loaded foot-ings." J. Soil Mech. and Found. Engrg.
Div., ASCE, 97(1), 95-117.
Purkayastha, R. D., and Char, A. N. R. (1977). "Stability
analysis of eccentrically loaded footings." J. Geotech. Engrg.
Div., ASCE, 103(6), 647-651.
Purkayastha, R. D. (1978). "Investigation of footing under
eccentric load." /. Indian Geotech. Society, New Delhi, India, 9,
220-234.
Saran, S. (1969). "Bearing capacity of footings subjected to
moments," thesis pre-sented to the University of Roorkee, at
Roorkee, India, in partial fulfillment of the requirements for the
degree of Doctor of Philosophy.
Saran, S. (1970). "Fundamental fallacy in analysis of bearing
capacity of soil." J. Inst, of Engineers, Calcutta, India, 50,
224-226.
Saran, S. (1971). "Bearing capacity of footings under inclined
loads." seminar on foundation problems. Indian Geotechnical
Society, New Delhi, India, Vol. II, IV-5.
Saran, S., Prakash, S., and Murthy, A. V. S. R. (1971). "Bearing
capacity of footings under inclined loads." J. Soil Mech. and
Found. Engrg., Japanese Society Soil Mech. and Found. Engrg.,
11(1), 47.
Saran, S., and Niyogi, B. P. G. (1970). "A model study of
footings subjected to eccentric inclined load in case of
cohesionless soil." Symp. on Shallow Found., Sarita Prakashan,
Meerut, India, Vol. 1, 29-35.
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Schultz, E. (1952). "Der Widerstand des Baugrundes gegen Schrage
Sohlpressungen" (in German). Die Bautechmik, 29(12), 336.
Selig, E. T., and Mckee, K. E. (1961). "Static and dynamic
behavior of small footings." J. Soil Mech. and Found. Engrg. Div.,
ASCE, 87(6), 29-47.
Sokolovski, V. V. (1965). Statics of grannular media. Pergamon
Press, New York, N.Y.
Terzaghi, K. (1943). Theoretical soil mechanics. John Wiley and
Sons, Inc., New York, N.Y.
Vesic, A. S. (1970). "Bearing capacity of shallow foundations."
Foundation engi-neering hand book. H. F. Winter Korn and H. Y.
Fang, eds., Van Nostrand, New York, N.Y., 121-147.
Zaharescu, E. (1961). "The eccentricity sense influence of
inclined loads on bearing capacity of rigid foundations." /. Nat.
Building Organisation, 6, 282-290.
APPENDIX II. NOTATIONS
The following symbols are used in this paper: B = width of
footing (L); C = cohesive force along DE (Fig. 3);
Ca = cohesion force along BE (Fig. 1) (F); C'a = cohesion force
along A'E (Fig. 1) (F) ;
c = unit cohesion (FL~ 2 ) ; Df = depth of foundation (L);
e = eccentricity (L); F = resultant of normal and frictional
force (F); / = load inclination with vertical (degree);
M = applied moment (FL); M = moment of weight force Ppr
(FL);
Mc = moment of cohesive force Ppc (FL); Mq = moment of surcharge
force Ppq (FL);
Mlc = moment of force Pc (FL) ; M2c moment of cohesive force C
(FL); Mlq = moment of force Pq (FL) ; M2q = moment of surcharge
weight Wq (FL); Mly = moment of force Pr (FL) ; M i , = moment of
weight W (FL);
m = mobilization factor; N = bearing capacity factor for weight
part ;
7VC = bearing capacity factor for cohesive part; Nq = bearing
capacity factor for surcharge part ; Pc = passive earth pressure
due to cohesion on soil mass CHD (Fig.
l ) (F) ; Pm = mobilized passive pressure for weight part (F)
;
Pmc = mobilized passive pressure for cohesion part (F) ; Pmq =
mobilized passive pressure for surcharge part (F);
Pp = passive pressure for weight part (Fig. 3) (F) ; Ppc =
passive pressure for cohesive part (Fig. 3) (F); Ppq = passive
pressure for surcharge part (Fig. 3) (F) ; Pq = passive earth
pressure due to surcharge on CH (Fig. 2) (F) ; Py = passive earth
pressure due to weight of soil CHD (Fig. 1) (F) ; Qd = total load
(F);
q = load intensity (FL - 2); 1689
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ra = initial radius of log spiral (i.e., BE) (L); x = (sin
a2)/[sin(a! + a 2 ) ] ;
xt = ratio of effective width of full width of footing y = (sin
a1)/[sin(a1 + a 2 ) ] ;
a, , a2 = wedge angles y = unit weight of soil (FL~3);
! = log-spiral angle on eccentricity side; cr = stress (FL~ 2) ;
()) = angle of internal friction; and
, = mobilized angle of internal friction.
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