Aegon would in return participate in the renewing of the
square. The landscape architects "West 8" from Rotterdam
were asked to make a design for the town square.
A part of
their plan was to build six so-called "follies". These follies are
large sails made out of gauze, stretched between steel cables
and steel tubes. The
gauze
will be overgrown
with
vegetation in time.
The follies vary from a simple single mast with several stays to
real tensegrity structures. A tensegrity can be described as an
island of compression in a sea of tension. The element under
compression (the tube) is held in place by cables. Each end
of the tube must be secured by at least three cables under
tension. The last word is very important: tension is needed to
give the structure stiffness. Therefore the structure will have
to be prestressed. If in a certain loadcase tension lacks in one
of the cables, the tube end becomes instable. Some of the
cables will lose there tension in some loadcases, but as long
as three tensed cables on each tube end remain this is no
problem for the structure. (The three tensed cables must also
go in different directions: if one sees the tube from above,
the cables must cover more than 180°).
Use of ESA-Prima Win
Technical questions in ESA-Prima Win:
The design process took quit some time. In this time several
versions of ESA-Prima Win were used. The last version was
3.40. The gauze
was not calculated, only the cables, the
tubes and the foundation.
Some of the specific problems of calculating the follies are
described below.
The structure needs tension to function. This is not only
valid for the real structure, but also for the calculation itself.
If a structure is not stiff, the stiffness matrix will have a zero
on the main diagonal. This means the equations cannot be
solved. By introducing prestressing the zero on the main
diagonal
will be replaced by a value representing the initial
strain in. This means the equations can be solved.
The structure behaves non linear.
With each (large)
deformation the stiffness of the structure changes.
With
the
Newton Rhapson procedure the stiffness
matrix is
recalculated after every iteration. This means that several
iterations are necessary, each time with a slightly different
stiffness
matrix. The calculation is stopped
when the
structure reaches equilibrium.
With some follies the load
had to be applied in steps (increments), otherwise the
deformations per iteration
would become to great. The
use of increments can help to keep the calculation stable.
In some follies certain cables would lose there prestressing.
As long as enough cables under tension per tube end
remain this is no problem. The cable element is removed
from the calculation
when the element comes under
compression.. If the cable is necessary for the stability of
the structure, the calculation will stop. The structure is no
longer stiff. This means that the level of prestressing has to
be raised or that extra cables have to be added.
With non-linear calculations superposing different load
cases in post processing is not possible. The load is
necessary to find the right equilibrium
with the right
structural stiffness. This means that non-linear load cases
have to be made. The following loads were used:
wind in
three directions (x, y and z direction: +, 0 and -), permanent
loads (+) en temperature loads (+, 0 and -). These loads can
be combined in 3x3x3x1x3=81 load cases. To reduce the
calculation time the number of load cases has to be
minimized. This is done by first looking at load cases with
the permanent load and only one variable load (load cases
1 to 10). The effect of this particular variable load on the
maximum and minimum stress in the cables and on the
foundation was studied. Then the worst variable loads for
the structures were combined in some extra load cases.
Experience with ESA-Prima Win when realising the
project:
The program
makes it possible to calculate a tensegrity
structure.
It was not necessary to model the gauze sails. I'
m not sure if
the sails could have been
modelled using plate elements
with the Newton Rhapson procedure for large deformations.
It is possible to
model the sails by replacing them by
numerous parallel cables. This will however greatly increase
the calculation time. Removing the sails made the calculation
faster, it sadly means that the loads on the cables had to be
inputted by hand. I have simplified this by only looking at the
projections of the sails in the three directions, not by
calculating the proper direction perpendicular to the sail for
each element.
In the first
models in ESA-Prima Win I encountered some
problems with the elements used. For example the output
showed large shear forces in the cables,
which could not be
real. In a later model I found out that the rotations at the
cable ends could no be correct.
All these problems
were solved in later patches or new
releases after having consulted SCIA. The models are very
sensitive to changes. I found that the outcome of the
calculations could vary when using different versions of ESA-
Prima
Win. In version 3.40 I suddenly had to use
more
increments to keep the calculation stable. The calculation
engine of ESA-Prima Win had changes a bit,
which
was
noticeable in the results.
Conclusion:
With these kinds of complicated and delicate structures one
has to be very careful using a powerful tool such as ESA-
Prima Win. You have to know what you want to find and
what you expect to find. Calculation by hand is hardly
possible, so when in doubt it is best to try to simplify the
model to a level
where you can understand and approve the
results.
Off course this is valid for all
models in ESA-Prima Win,
but it is specially important for the models were checking the
results by hand is not possible.
The work on the follies has been completed in April this year.
During erection the stresses in the structure have been
measured to check the validity of the calculations.
The measurements confirmed the calculations.
Now we only
have to wait for the vegetation to grow on the gauze.
Modules used:
3d frame
Dynamic document
2nd order frame
Physical non linear conditions
13
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