Impedances
The impedances relate displacements/rotations with forces/moments. In general,
this can be written as a matrix relation , where
is the impedance
matrix,
is the displacements/rotations vector, and
is the forces/moments
vector. Each impedance term is a frequency dependent complex-valued function
which can be written as:
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where is the real part or stiffness coefficient,
is the dimensionless frequency,
is the circular frequency (rad/s),
is a pile diamater,
is a shear wave velocity, and
is
the damping coefficient.
On the other hand, for pile configurations with planes of symmetry calculations
can be made applying symmetry conditions, so the studied geometry is reduced.
This way, the results are obtained more quickly and easily: if there are
planes of symmetry, the studied geometry is reduced to
of the original.
Vibration modes
The impedances recieve different names depending on the displacement/rotation
they relate. PILEDYN does not take into account torsion, therefore it considers
five vibration modes. Those are:
VM 1: displacement along X axis
To produce this displacement (), a force applied along x axis
(
) is needed. Nevertheless, as a consecuence of
and the imposition
of null rotation, a resistant momentum appears (
).
Then, two impedances can be obtained with this vibration mode: Kxx,
the horizontal impedance along x axis, and Kryx, the
crossed impedance along y axis that appears with .
VM 2: displacement along Y axis
In this case, the situation is similar to the one described in Vibration
mode 1, but changing the direction in which the pile vibrates to the y axis.
The diagram below represents it:
Where is the displacement along y axis,
the force
applied ncessary to produce
and
(negative) is the resistant
momentum.
Then, two impedances can be obtained with this vibration mode too:
Kyy, the horizontal impedance along y axis, and
Krxy, the crossed impendance along x axis that appears
with .
VM 3: displacement along Z axis
Now, to produce the displacement along z axis () a force applied
in the same direction (
) is needed.
Then, only one impedance can be obtained with this vibration mode:
Kzz, the vertical impedance.
VM 4: rotation along X axis
To produce a rotation along x axis (), a momentum
have to be applied along x axis. However, as a consecuence of
,
a resistant force appears along z axis (
), that
produces another momentum. This can be seen in the diagram below:
Then, two impedances can be obtained with this vibration mode:
Krx, the rocking impedance in x axis, and Kyrx,
the crossed impedance that appears with .
VM 5: rotation along Y axis
In this case, another momentum is needed (), applied
along y axis. But, as a consecuence of
and again the
imposition of null desplacement and rotation in other axes, a resistant force
appears along z axis (
), that applied in x axis
produces another momentum. This can be seen in the diagram below:
Then, two impedances can be obtained with this vibration mode too:
Kry, the rocking impedance along y axis, and Kxry,
the crossed impedance along y axis that appears with .
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