twice the effective shear modulus, G,
the footing by the moment of inertia of the footing in
determined in the geotechnical investigation. The
the direction of loading. In general, however, the
lower-bound stiffness should be based on one-half
uniformly
distributed
vertical
and
rotational
the effective shear modulus; thus, the range of
stiffnesses are not equal. The two may be effectively
stiffness should incorporate a factor of four from
decoupled for a Winkler model using a procedure
lower- to upper-bound.
Most shallow bearing
similar to that illustrated in Figure 9-5. The ends of
footings are stiff relative to the soil upon which they
the rectangular footing are represented by End Zones
rest. For simplified analyses, an uncoupled spring
of relatively high stiffness, with overall length of
model, as shown in Figure 9-1b, may be sufficient.
approximately one-sixth of the footing width. The
The three equivalent spring constants may be
stiffness per unit length in these End Zones is based
on the vertical stiffness of a B x B/6 isolated footing.
determined using conventional theoretical solutions
for rigid plates resting on a semi-infinite elastic
The stiffness per unit length in the Middle Zone is
medium.
Although frequency-dependent solutions
equivalent to that of an infinitely long strip of
are available, results are reasonably insensitive to
footing.
In some instances, the stiffness of the
loading frequencies within the range of parameters
structural components of the footing may be
of interest for buildings subjected to earthquakes. It
relatively flexible compared to the soil material. A
is sufficient to use static stiffnesses as representative
slender grade beam resting on stiff soil is an
of repeated loading conditions. Figure 9-2 presents
example. Classical solutions for beams on elastic
stiffness solutions for rectangular plates in terms of
supports can provide guidance regarding when such
an equivalent circular radius.
Stiffnesses are
effects are important. For example, a grade beam
supporting point loads spaced at a distance of L
adjusted for shape and depth using factors similar to
those in Figure 9-3.
For the case of horizontal
might be considered flexible if:
translation, the solution represents mobilization of
EI
base traction (friction) only.
If the sides of the
< 10k sv B
(9-8)
L4
footing are in close contact with adjacent in situ
foundation soil or well-compacted fill, significant
additional stiffness may be assumed from passive
where, for the grade beam,
pressure. A solution for passive pressure stiffness is
presented in Figure 9-4. For more complex analyses,
a finite element representation of linear or nonlinear
foundation behavior may be accomplished using
I = moment of inertia
Winkler component models.
Distributed vertical
stiffness properties may be calculated by dividing the
B = width.
total vertical stiffness by the area.
Similarly, the
uniformly distributed rotational stiffness can be
For most flexible foundation systems, the unit
calculated by dividing the total rotational stiffness of
subgrade spring coefficient, ksv, may be taken as
9-4
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