Geoscience Reference
In-Depth Information
We are now in a position to compare the rotational and vertical stiffness of the founda-
tion on the nonlinear soil. Because of the nonlinearity this is not as simple as the ratio
plotted in Figure 10.10, now it depends on the vertical load applied to the foundation.
Presented in Figure 10.12b is the ratio of rotational to vertical stiffness at various points
aroundtheaveragebearingpressure-settlementcurve(Figure10.12a),frominitialpoints
with “elastic” behaviour to points approaching bearing failure. That is we evaluate, for
valuesoftheverticalloadonthefooting,therotationalstiffnessfortheinitialapplication
ofmomentwiththeverticalloadheldconstant andtheverticalstiffnessforanadditional
vertical load increment with no moment. This ratio is found to decrease as the load on
the footing increases, or in other words as nonlinear soil behaviour becomes more sig-
nificant. Therefore as the footing load increases the stiffnesses of the footing decrease
but the decrease in the rotational stiffness is more rapid than the decrease in vertical
stiffness.
ItwascommentedearlierthatthedistributionofbearingpressureshowninFigure10.13b
is not linear, but the peak at the edge is not as severe as that for the continuous elastic
model. The decreasing stiffness ratio plotted in Figure 10.12b indicates that as bearing
failureofthefootingisapproacheditispossiblethatthebedofspringsmodelrepresents
the rotational stiffnessof the footing more effectively.
The above discussion shows that when considering foundation stiffness the simple rep-
resentation of the soil beneath a shallow foundation as a bed of springs is unlikely to be
fully satisfactory. When the soil behaves as an “elastic” material (a common representa-
tionwhentheloadsonthefoundationareconsiderablylessthanthebearingstrength)the
bed of springs, calibrated so that the vertical stiffness is correct, under predicts the rota-
tionalstiffnessofthefoundation.Wefoundthatwhennonlinearsoilbehaviouroccursthe
nonlinearity has a more significant effect on the rotational than on the vertical stiffness.
Consequently,nonlinearsoilbehaviourappearstoleadtoanimprovementasthestiffness
ratio decreases with increasing nonlinearity, Figure 10.12b. However, even then the sim-
plespringapproach,withnonlinearsprings,isnotfullysatisfactoryasthepressureatthe
footing edge isstillnot zerowhen thedisplacement iszero, Figure 10.13b.
Despite these difficulties it may be possible to get the approximately correct stiffness
behaviour for the footing if a bed of nonlinear springs is coupled with a nonlinear rota-
tional spring at the centre of the footing. This means that the correct stiffness ratio could
be achieved in the “elastic” region. Like the linear springs this spring would need to be
nonlinear with a decreasing rotational stiffness as the rotation increases. In this way it
might be possible todevelop asuitable nonlinear spring configuration that represents the
foundation stiffness. However, even if this is achieved the model would not give the cor-
rect bending moment and shear force distribution in the footing. So obtaining a valid
stiffness model for a shallow foundation is fraught with difficulties, but certainly more
than a simplebed ofsprings is needed.
