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for rigid structural elements. The basis for these comparative comments is, of course,
response of acontinuous elastic medium.
The main thrust of this section of the paper is concerned with the vertical and rotational
stiffness of shallow foundations. The stiffnesses obtained using the two models are com-
pared.Themainconclusionisthat,forgivendimensionsofarigidrectangularfoundation
restingonthegroundsurface,therotationalstiffnessofabedofspringsislessthanthatof
thesamefoundationonacontinuouselasticmaterial.Furthermore,whenthesoilismod-
elledasanonlinearmaterialtheratioofrotationaltoverticalstiffnessgraduallydecreases
as bearing failureis approached.
4.1. SHALLOW FOUNDATIONS ON ACONTINUOUS ELASTIC SOIL
Gazetas (1991) presents a set of expressions, developed by him and his co-workers, giv-
ing vertical and rotational stiffness of rigid shallow foundations of arbitrary shape and
embedment condition.Atthispointweconsideronlyfoundationsonthegroundsurface.
The vertical stiffnessof a rectangular foundation on abed of springs is given by
K
V
springs
=
A
b
k
s
(10.4)
where:
K
V
springs
is the vertical stiffness of the foundation on a bed of springs and
k
s
is the spring stiffness per unit area of the foundation and
A
b
is the contact area of the
foundation.
The rotational stiffness of the rectangular foundation on abed of springs isgiven by
N
L
springs
/
2
1
2
A
fs
k
s
s
2
2
K
=
(
2
j
−
1
)
(10.5)
θ
springs
j
=
1
where:
k
s
is thevertical stiffnessof the bed of springs per unit area of the foundation,
s
is the spring spacing (assumed tobe the same in thelength and breadth directions),
j
is a counter,
N
L
springs
isthenumberofrowsofsprings(assumedtobeeven)inthelongitudinaldirec-
tion of thefooting
,
and
A
fs
istheareaofthefoundationassociatedwitheachrowofsprings
(
=
L
/
s
)
(
=
B
×
s
)
.
InFigure10.10theratiooftherotationalstiffnesstotheverticalstiffnessofsquarefoun-
dations resting on the ground surface is plotted for both the elastic soil and the bed
of springs. These calculations were done by setting the spring stiffness for the bed of
springs so that, for a given foundation dimension, the vertical stiffness was the same
for both models. Figure 10.10 makes very clear that the rotational stiffness of a shallow
