Eccentric Loading
Eccentric Loading may result from a load applied off the center of the footing or
from a concentric load plus a bending moment. For the purpose of determining
under the footing the pressure under the footing the moment may be removed by
shifting the vertical load to a fictitious location with an eccentricity
e=momento/vertical load. In the analysis of an eccentrically loaded footing two
separate problems are confronted:
1.For the purpose of structural desing, the pressure against the bottom of the
footing, commonly called contact pressure, is assumed to have a planar
distribution. When the load is applied within the kern of the footing area, common
flexural formulae are applicable.
Q Mx M
q= ± x± y y
A Iy Ix
When ex, ey or eb, el excced a certain limit, Eq (6-9) or (6-9a) gives a negative
value of q which indicates tension between the soil abd bottom of footing. Unless
the footing is werghted down by surcharge loads, the soil cannot be relied upon for
bonding to the footing and offering tensile resistance. Therefore, the flexual
formulae (Eq 6-9) an (6-9a) are applicable only when the load is applied within a
limited area which is know as the kern and is shown shaded in Fig 6-14 a
The procedure for determination of soil pressure when the load is applied outside
the kern is simple in principle but laborious. Cases for rectangular and circular
footing have been worked out and the kerns are shown by shaded areas in fig 6-14
ayc
For footing of other shapes, the graphical method of successive trials is probably
the simplest for practical solutions.
The graphical method, similar to any other method, is based on the assumption
that the pressure varies linearly with the distance to the neutral axis from zero at
the neutral axis to a maximum at the most remote point and on the requirement of
statical equilibrium that the resultant of the soil pressure should lie on the line of
action of the applied load Q.
The procedure is as follows. Draw a trial neutral axis N-N, Fig 6-14 (b) and a line
ab perpendicular to N-N is under compression while the area on the other side of
N-N is unstressed. The intensity of stress at a given point varies in simple
proportion with its perpendicular distance from N-N. The compression area is
divided into several narrow strips of uniform width dy, rumning parallel to N-N.
The unit pressure acting on this strip is equal to (Y,X)qb, where qb is the unit
pressure at point b, and the total pressure is equal to (Y,X)qbIdy. The total
pressure may be represented by the shaded strip with a length of (Y,X)l. This
shaded strip, if under a uniform pressure qb, carries the same load as the whole
strip under the actual pressure (Y,X)qb. Therefore, it may be called a transformed
strip. All the transformed strips form a transformed area. If the location of the trial
neutral axis N-N is correct, the centroid of the transformed area will coincide with
the point of action of the load Q. For practical purposes, the centroid or center of
gravity of the transformed area may be determined by cutting out a cardboard of
the same shape and balancing the board on a pencil point. The cardboard will
balance only when it is supported on the center of gravity. Several such trials will
enable he engineer to approach the correct location of the neutral axis.
For determination of ultimate or allowable bearing capacity of an eccentrically
loaded footing, the concept of useful width has been introduced. By this concept,
the portion of the footing which is symmetrical about the load is considered useful
and the other portion is simply assumed superfluous shown in Fig 6-15, the useful
widths are B-2eb and L-2el, the equivalent area (B-2eb)(L-2el) is considered as
subjected to a central load for determination of bearing capacity.
The concept above simply means that the bearing capacity of a footing decreases
linearly with the eccentrically of load, as is shown by a straight line in Fig 6-16. In
cohesive soils, this linear relationship prevails, but in granular soil, however, the
reduction is parabolic rather than linear.
Therefore, the reduction factor shown in Fig 6-16 should be used for design
purposes: First the bearing capacity of the footing is determined on the basis that
the load is applied at the centroid of the footing. Then, this bearing capacity is
corrected by multiplying with the factor shown in fi 6-16
