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

base traction (friction) only.

If the sides of the

< 10*k * sv B

(9-8)

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

Winkler component models.

Distributed vertical

stiffness properties may be calculated by dividing the

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, *k*sv, may be taken as

9-4

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