Example (7.2): Rectangular foundation subjected to eccentric loading
Introduction
The simplest model for determination of the contact pressure
under the foundation assumes a planar distribution of contact pressure on the bottom of the
foundation (statically determined). In which the resultant of soil reactions coincides with the
resultant of applied loads. If all contact pressures are compression, the foundation system
will be considered as linear and the contact pressures in this case is given
directly.
If the foundation subjects to big eccentricity, there will
be negative contact pressures on some nodes on the foundation. Since the soil cannot resist
negative stress, the foundation system becomes nonlinear and a resolution must be carried out to
find the nonlinear contact pressures.
The nonlinear analysis of foundation for the simple
assumption model has been treated by many authors since a long time, where several analytical and
graphical methods were available for the solution of this problem.
Pohl (1918) presented
a table to determine the maximum corner pressure
max qo
for arbitrary positions of the resultant
N.
Hülsdünker
(1964) developed a diagram using the numerical
values of this table from
Pohl (1918) to determine the
maximum corner pressure max qo.
For one corner detached footing, the closed form formulae cannot be
used. Therefore, Pohl
(1918) and
Mohr (1918) proposed a
method to estimate the neutral axis through the trial and error. Besides tables and diagrams,
Graßhoff
(1978) introduced also influence line charts can
be used to determine the contact pressure ordinates.
Peck/ Hanson/ Thornburn
(1974) indicated a trial and error method to
obtain the neutral axis position for rectangular footing subjected to moments about
both axes. Jarquio/ Jarquio
(1983) proposed a direct method of proportioning a rectangular
footing area subjected to biaxial bending. Irles/
Irles (1994) presented an analytical
solution for rectangular footings with biaxial bending, which will lead to obtain explicit solutions for
the corner pressures and neutral axis location.
The determination of the actual contact area and the maximum
corner pressure max qo
under eccentric loaded foundation with irregular shape is very
important. For Tshape foundation that is loaded eccentrically in the symmetry axis,
Kirschbaum
(1970) derived formulae to determine the maximum corner pressure
max qo.
For some foundation areas with polygonal boundaries,
Dimitrov (1977) gave
formulae to determine the foundation kern and corner pressure
max qo. For the same purpose,
Miklos
(1964) developed diagrams. For general cases of
foundation,
Opladen (1958)
presented graphical procedure.
Most of the analytical methods used to determine the contact
area and corner pressures for eccentric loaded foundations are focused on regular
foundations where irregular foundations can be analyzed only by graphical procedures. In this paper, an
iteration procedure is presented to deal with nonlinear analysis of foundations for simple
assumption model. The procedure can be applied for any arbitrary foundation shape and is suitable
for computer programs. The following section describes this procedure.
Description of the problem
For comparison with complex foundation shape, no analytical
solution is yet available. Therefore, for judgment on the nonlinear analysis of
foundations for simple assumption model, consider the rectangular foundation shown in Figure (7.5).
The foundation has the length L
= 8.0 [m] and the width
B
= 6.0 [m]. The foundation carries an eccentric
load of N
= 2000 [kN].
Both of the xaxis
and yaxis
are main axes, which intersect in the center of gravity of the foundation area
s. The position of resultant
N
is defined by the ordinates
x
= ex
and
y
= ey.
Within the rectangle foundation area five zones are
represented. It is found that, the contact area and maximum corner pressure
max qo
depending on the position of the
resultant N
in these five zones (Irles/
Irles (1994)). In this example, the
maximum corner pressure max qo
is obtained using the program ELPLA for each zone and compared with
other analytical salutations, which are available for rectangular foundation.
