DIMENSIONING FOOTINGS SUBJECTED
TO ECCENTRIC LOADS
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By William H. Highter, 1 M. ASCE and John C. Anders 2
INTRODUCTION
Eccentric loading of shallow foundations occurs w h e n a vertical load
is applied at a location other than the centroid of a footing, or a footing
is subjected to a m o m e n t as well as a concentric vertical load. In the
latter case, the vertical load can be transferred to a n eccentric position
defined by
M.
eb = —
,
and
ei =
Mb
~p~
W
in which P = vertical load; Mh and M i = m o m e n t s about the short and
long axes of a rectangular footing, respectively; and eh a n d ex = eccentricities of the load P about the centroid of the footing in the direction
of the short and long axes, respectively.
Analysis of an eccentrically loaded footing requires consideration of
the structural design a n d bearing capacity. The approach usually used
in structural design is to assume that the contact pressure at the bottom
of the footing is planar. If the eccentric loading is within the kern of the
footing, the contact pressure distribution can be obtained from the flexural formula. If, however, t h e eccentric loading is outside the kern, a
trial and error graphical procedure m u s t be used, based on: (1) The assumption that the contact pressure varies linearly from zero at the neutral axis to a maximum at a point farthest from the neutral axis; a n d (2)
the requirement that t h e resultant of the soil contact pressure m u s t coincide with the point of application of the applied load, P. Teng (5) p u b lished a table (for circular footings) and a figure (for rectangular footings)
that gave factors from which the maximum contact stress could be calculated. Even with this, the structural design of footings subjected to
eccentric load is an iterative procedure. More recently, Jarquio and Jarquio (1) presented a direct m e t h o d of designing a rectangular footing
w h e n the maximum contact pressure is specified and the contact stresses
are compressional everywhere beneath the footing, i.e., the flexural formula is applicable.
The approach often used to determine the bearing capacity of a footing
subjected to an eccentric loading is based on Meyerhof's (2) observation
that "at the ultimate bearing capacity of the foundation the distribution
of contact pressure is not even approximately linear, and a very simple
solution of the problem is obtained by assuming that the contact pres'Prof. of Civ. Engrg., Univ. of Tennessee, Knoxville, Tenn. 37996.
Geotechnical Engr., Soil and Material Engrs., Inc., Blountville, Tenn. 37617.
Note.—Discussion open until October 1, 1985. 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
March 27, 1984. This paper is part of the Journal of Geotechnical Engineering,
Vol. I l l , No. 5, May, 1985. ©ASCE, ISSN 0733-9410/85/0005-0659/$01.00. Paper
No. 19696.
2
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J. Geotech. Engrg. 1985.111:659-665.
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sure is identical to that . . . for a centrally loaded foundation but of
reduced width." Meyerhof suggested that when there is two-way eccentricity, the procedure be extended "by finding the minimum effective
contact area . . . such that its centroid coincides with that of the load."
Meyerhof's approach is preferable to an alternative method of determining bearing capacity of eccentrically loaded footings which considers
maximum contact pressure because experimental evidence supports the
concept of a reduced, effective area; furthermore, Meyerhof's experiments showed that the alternative method gives rather conservative results for clays. For sands, the results are "reasonable for small eccentricities but unsafe for greater eccentricities."
Having calculated the reduced or effective area, A', of a footing subjected to an eccentric load using the Meyerhof procedure, and defining
L' as the longest dimension of the effective area, the effective width, B',
is then calculated from
B
A'
(2)
'=F
The effective area is thus assumed to be rectangular.
In analysis, these effective dimensions are then used in the well-known
bearing capacity equation (3,4) to determine the allowable load, Q, on
the footing:
/
B'
\A'
Q = \kcdcicCNc + X^d^yDfN,, + \ydyiy — yNyJ —
(3)
in which FS = an appropriate factor of safety; Nc, Nq and Ny = bearing
capacity factors for a continuous footing; 7 = unit effective weight of the
soil; C = undrained shear strength of the soil; \ c , X, and \y = shape
factors used for other than continuous footings; dc, dq and dy = factors
used to account for the increase in bearing capacity due to the strength
of the soil above the foundation level; ic, /', and iy = factors used to
account for the inclination of a concentric load; and Df = minimum distance from ground level to the bottom of the footing. Note that since
the last term in Eq. 3 (which contains B') vanishes for cohesive soil (Ny
= 0) but not for cohesionless soils, the effect of eccentricity is more pronounced for footings on sands than on clays.
In design, determining the bearing capacity of a footing from Eq. 3 is
always an iterative procedure because, while the design objective is to
dimension the footing, the shape factors depend on the relative dimensions of the footing. When the loading is eccentric, the design is more
cumbersome because the process of finding the reduced effective width
and area of a footing subjected to an eccentric load is time-consuming.
The eccentricity of the applied load is easily determined from vertical
load and moment data, but since determining the dimensions of the
footing is the purpose of the design procedure, the reduced (effective)
dimensions cannot be found directly. To aid the engineer, normalized
design charts for determining effective dimensions have been prepared
and are presented here. The design process necessarily remains iterative, but the design charts will enable the engineer to design circular or
660
J. Geotech. Engrg. 1985.111:659-665.
rectangular footings subjected to eccentric loads much more easily and
faster than was previously possible.
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ONE-WAY ECCENTRICITY
For rectangular footings subjected to one-way eccentricity, where the
load is applied along a line joining the midpoints of opposite sides of
the footing, the effective foundation dimensions are
B' = B-2eb
and
and
L'= L when
B' = B and
L' = L - 2et
ex = 0 [Fig. 1(a)]
when
(4)
eb = 0 [Fig. 1(b)]
(5)
Because of symmetry, eccentric loading on a circular footing is always
one-way. Normalized solutions for A'/R2 and B'/R are shown in Fig. 2
as a function of Er/R in which E, = eccentricity of the load and R =
radius of the footing. To use this graphical solution, enter Fig. 2 with
the eccentricity Er (calculated from load and moment data) and a known
footing radius. The effective dimensions A' and B' can then be determined and used in the bearing capacity equation to analyze the bearing
?3J
2«l
(W.
FIG. 1.—(a) Single Eccentricity (ex = 0); (ft) Single Eccentricity (eb = 0)
•
H
ff
1 n
\
\
\
R*
"
\
\
\
\
1.0
\
\
A1
R!
\
B'
\
i
i
L'
\
\ s
\
erv/A
s
\
\ N
\,
0.0
0.0
0.1
02
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
FIG. 2.—Normalized Effective Dimensions for Circular Footing
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J. Geotech. Engrg. 1985.111:659-665.
capacity of the footing. In design, the procedure is iterative and radii
are assumed until a footing having an acceptable safety factor is obtained.
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TWO-WAY ECCENTRICITY
When the eccentricity is in both directions, the problem is more complex because the designer does not know the shape of the effective area.
For rectangular footings, there are four possible cases, depending on the
magnitudes of ei/L and eb/B. In developing these figures, the eccentric
load was assumed to act in the first quadrant of the rectangular footing.
Case 1.—In Case 1, e-^/L S 1/6 and eb/B g 1/6, and the effective area
is triangular (Fig. 3). The normalized dimensions of the effective area
can be calculated directly:
B1_1.5-3e„
B
B
L i = L5_-_3e i
L
L
The effective area A' can be calculated from Bi and L j . Letting L'
equal the larger of Bx and Li, B' is then calculated using Eq. 2. In lieu
of Eq. 6, the uppermost curves of Figs. 4 and 5 can be used to find
Li/L as a function of ej/L, and B1/B as a function of eb/B, respectively,
for Case 1.
Case 2.—In Case 2, eb/B < 1/6 and 0 < ex/L < 0.5. Entering Fig. 4
with e : /L and eb/B, locate et/L on the ordinate and move to the right
until the first radial line corresponding to eb/B is encountered. The abscissa value corresponding to this point is L2/L. Moving further to the
right across the broken line, a second radial line corresponding to eb/B
will be found. The value on the abscissa corresponding to this point is
Lx/L. The effective area A' can then be calculated knowing L-i and L 2 ;
L' is the larger of Lx and L 2 , and B' is found from Eq. 2.
Case 3.—In Case 3, e%/L < 1/6 and 0 < eb/B < 0.5. this case is similar
\
X//®-
l
/.
B
FIG. 3.—Case 1: Effective Area (eb/B g 1/6 and e,/L £ 1/6)
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J. Geotech. Engrg. 1985.111:659-665.
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FIG. 4.—Case 2: Normalized Effective Dimensions (and LJL for Case 1)
FIG. 5.—Case 3: Normalized Effective Dimensions (and BJB for Case 1)
to Case 2 except that the base of the trapezoidal effective area is L (Fig.
5). Bx and B2 are found from Fig. 5; A' is then calculated knowing Bx
and B 2 ; L' = L; and B' is found from Eq. 2.
Case 4.—In Case 4, eb/B < 1/6 and eJL < 1/6. In Fig. 6, the family
of ei/L curves sloping upward to the right represents values of B2/B on
the abscissa, while the family of ex/L curves sloping downward to the
right represents values of L2/L. A\ is calculated from B2 and L 2 ; L' = L
in Case 4; and B' is determined from Eq. 2.
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J. Geotech. Engrg. 1985.111:659-665.
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0,00 I 1 1—I 1 1—I 1—I '—I 1 1 1 1—I 1 1—I 1—I '—I—I—•—I—I—I
0.0
0.1 0.2 0.3
0.4 0.5
0.6 0.7 0.8 0.9
1.0
Be
L,
B ' L
1
FIG. 6.—Case 4: Normalized Effective Dimensions
In practice, it is not necessary to distinguish between Cases 2, 3 and
4 before using Figs. 4-6 because each figure represents unique combinations of eb/B and ex/L.
COMPUTER PROGRAM
A computer program has been written to solve the problem of eccentrically loaded circular footings and rectangular footings with twoway eccentricity. Using the program is much faster than using the charts
(Figs. 4-6). The program prompts the user to specify a circular (C) or
rectangular (R) footing. Any other symbol stops the problem. If C is
selected, the program prompts for the eccentricity and radius Er and R,
respectively. Output includes the input data along with the effective area
A' and effective width B'.
It R is selected, the program prompts for the length and width of the
footing and the eccentricities. Output includes input data, the dimensions of the effective area, and the effective width of the footing.
Versions of the program hhave been written in BASIC for both the
IBM PC (DOS 2.1) and the Apple II + . Interested readers may obtain a
copy of the program by sending a disk along with a self-addressed
stamped mailer to the first writer. Be sure to specify either the Apple or
IBM version, and please initialize disks for the Apple II + .
APPENDIX.—REFERENCES
1. Jarquio, R., and Jarquio, V., "Design Footing Area with Biaxial Bending," Journal
of Geotechnical Engineering, ASCE, Vol. 109, No. 10, Oct., 1983, pp. 1337-1341.
2. Meyerhof, G. G., "The Bearing Capacity of Foundations Under Eccentric and
Inclined Loads," Proceedings of the Third International Conference on Soil Mechanics and Foundation Engineering, Vol. 1, 1953, pp. 440-445.
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3. Meyerhof, G. G., "Some Recent Research on the Bearing Capacity of Foundations," Canadian Geotechnical Journal, Vol. 1, No. 1, 1963, pp. 16-26.
4. Perloff, W. H., and Baron W., Soil Mechanics-Principles and Applications, Ronald
Press Company, New York, N.Y., 1976, 745 pp.
5. Teng, W. C., Foundation Design, Prentice-Hall, Inc., Englewood Cliffs, N.J.,
1962, 466 pp.
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