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Figure 4.27 The pull back moment-rotation curves from all the
snaps in test 7
Figure 4.28 The pull back moment-rotation curves from all the
snaps in test 9
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Rotation (millirads)
Moment(kNm)
Snap 1
Snap 2
Snap 3
Snap 4
Snap 5Snap 6
Snap 7
Snap 8
Snap 9
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Rotation (millirads)
Moment(kNm)
Snap 1
Snap 2
Snap 3
Snap 4
Snap 5
Snap 6
Snap 7
Snap 8
Snap 9
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Figure 4.29 Dynamic moment-rotation for test 7 snap back 4
Figure 4.30 Dynamic moment-rotation for test 7 snap back 5
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Rotation (millirads)
Moment(kNm)
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4.3.13.3
Curve Fitting
The static moment-rotation curves were fitted with curves using
Matlabs nonlinear
regression function that enables an equation to be fitted to an
arbitrary curve
(MathWorks 2009). Producing equations that represent the static
moment rotation
curves given in Figures 4.27 and 4.28 would allow a designer to
choose a rotational
stiffness based on what rotations are expected, and thus proceed
with designing for
rocking foundations. Sullivan et al. (2010) say this particular
topic needs to be
researched further, and the results from the static moment
rotation curves gives us an
insight into how much stiffness degradation takes place during
rocking.
Kondner (1963) gives an equation for a hyperbolic stress strain
relationship for cohesive
soils, and describes the factors contributing to the curve. The
form of the stress strain
curve is:
baQ
(4.7)
where Q= shear stress; = axial strain; and aand bare parameters
for the hyperbolic
curve, given below.
Figure 4.31 The stress strain curves that give the parameters a
and b. Actual curve left, transformed curve to estimate parameters
for stress strain behaviour right
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Although these parameters are for stress-strain behaviour they
can be applied to
moment-rotation behaviour as well. The parameter that gives the
initial slope of the line,
a, is related to the initial stiffness of the system, and the
parameter that specifies the
horizontal asymptote, b, relates to the moment capacity of the
system.
Figure 4.32 gives the static moment-rotation curves from test 7
(same as Figure 4.27)
along with upper and lower bounds from the curve fitting. Figure
4.33 gives a close up
of those upper and lower bounds compared with snap-backs 1 and 9
respectively. The
curves were created by having an initial part that was elastic,
and then applying the
nonlinear regression at some defined moment the point where
initial nonlinearity
occurs. As is evident the curve functions can capture the static
moment rotation
distribution that was recorded in the experiments. The initial
elastic part of the curve
was calculated as around 60% of the Gazetas formula for the
upper bound and around
20% for the lower bound. Subsequently the ultimate moment
capacities for each were
110 kNm for the upper bound and around 85 kNm for the lower
bound.
Figure 4.32 The upper and lower bound of the fitted curves, with
the 9 snap-backs oftest 7 plotted in grey.
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Figure 4.33 A close up with of the upper (left) and lower
(right) bound fitted curves
for test 7
Figure 4.34 The upper and lower bound of the fitted curves, with
the 9 snap-backs oftest 9 plotted in grey.
Figure 4.34 gives the upper and lower bounds to the curve
fitting for test 9 along with
all the snap-backs of that test. Again it shows that the
nonlinear behaviour can be
captured by the equation. The upper bound had an initial
stiffness of around 80% of the
Gazetas value and a moment capacity of 110 kNm the same as what
was found for test
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Rotation (millirads)
Moment(k
Nm)
Upper Bound
Lower Bound
