5.9
RECTANGULAR PLATE, PLATE NO. 19
We want to compute a thick
rectangular plate of steel.
Data:
We will draw the plate
super structure in AutoCAD. Draw
one single super element plate type No.20,
which will be subdivided by the mesher Z88N
into 8 x 8 = 64 plates of type No.19,
i.e. with 16 nodes each. Of course, for this
example you could use an editor and generate the mesh generator input
file by
hand at the same pace:
Windows: AutoCAD LT,
drawing the rectangular plate.
You'll find the exact
procedure plottet in chapter 2.7 -
however, try it by yourself and export the
drawing as Z88X.DXF into the Z88 directory. If it doesn't work at all
(but it
really does):
Copy B18_X.DXF into
Z88X.DXF
Windows: CAD converter
Z88X. Looks very similar on UNIX machines.
Choose from Z88X to
Z88NI.TXT. Then, launch the mesher Z88N:
Windows: mesh generator
Z88N. Looks very similar on UNIX machines.
Now you may look at the
structure with Z88O:
Windows: Plot program
Z88O, undeflected structure. Looks similar on UNIX computers.
Now you've got some work:
you must read off the node numbers for the boundary conditions in Z88O.
We have
to decide how to support the plate. We'll choose "cutting edges",
i.e. the boundaries are supported by a "bezel" above and below. This
allows angular movement crosswise to the bezels, but fixture in
direction of
the bezels.
If you want to support the
boundary in front, i.e. running in X direction, with cutting edges,
then you
must fix the degree of freedom 1 (the Z direction) and the degree of
freedom 3
(the rotation around the Y axis).
We've got 625 nodes in total. Which to support ? Good question ! In order to save some work (seldom a good idea) we'll try to fix only the corner nodes of the elements, which lay on the boundaries. This nodes are
Windows: read off the
the nodes with Z88O. Looks similar on UNIX machines.
See the beginning and the
end of the boundary conditions file Z88I2.TXT
(if you are too lazy to do the work
of entering the boundary conditions: B18_2ROU.TXT) :
68
1 1 2 0.
1 2 2 0.
1 3 2 0.
4 1 2 0.
4 2 2 0.
....
622 1 2
0.
622 2 2
0.
625 1 2
0.
625 2 2
0.
625 3 2
0.
We may now launch one of
the solvers. Because the structure is really tiny, the Cholesky solver is the right choice. The
displacement file Z88O2.TXT gives us the information for node 313,
which lies
exactly in the middle of the plate:
313 +1.1236511E+001 -2.1751298E-008
+2.1751298E-008
The deflection U2 (i.e. the
rotation around the X axis) and U3 (i.e. the rotation around the Y
axis) are
zero, looks good. The deflection U1, i.e. w, is 11,24 mm.
"Analytically" (this is also only an approximation for thin plates,
ref. to the classical mechanics literature) one computes:
f=
(0.71 * p * b4)
/ (E * h3)=
(0,71 * 46,42 * 5004)
/ (206.000 * 1003)
= 10 mm
This results in a variety
of (10 - 11,24) / 10 * 100 = 12%.
Here's why. Firstly, the
analytical formulae in the literature are thin plates of the Kirchhoff
type
neglecting the shear forces, secondly, this formulae were won with
series
expansion and thirdly, we could truely put some more work into a better
formulation of the boundary conditions. Here's how our plot looks with
a
magnification factor of 50:
See how the boundaries
raise between the corner nodes? Guess we must swallow the bitter pill
and
support all the nodes laying on boundaries (copy file B18_2.TXT
to Z88I2.TXT). This results in:
w at node 313: 10,5 mm,
variety to the analytical calculation about 5
% (the
analytical calculation supplies thin plates and is not very exact here.
This
thin plates should feature a thickness of about 1/50, 1/100 or fewer of
the
main dimensions!)
We may calculate the
stresses "analytically":
Sigmax = Sigmay
= (1,15 * p * b2) / h2 = (1,15 * 46,42 * 5002)
/ 1002 = 1.335 N/mm2
The stress parameter file Z88I3.TXT needs the following entries for
computing the stresses in the corner nodes:
0 0 0
After running Z88D you
may read off the stresses of node
313 from the elemente 28, 29, 36 or 37; it is the node with XX= 600 and
YY=
600: Sigmax = Sigmay = 1.334 N/mm2 .
Now the
boundaries are supported properly.
Finally, we'll compute the
stresses in the Gauss points and, thus, adjust Z88I3.TXT as follows:
4 0 1
After a Z88D run
we may look at the von Mises
stresses:
Windows: Plot of the von Mises stresses in the 4 x 4 Gauss points. Z88O. Looks similar on UNIX machines.
Windows: Plot of the Z
displacements. Z88O. Looks similar on UNIX machines.
Now you've got a
small impression of plate calculation. Consult the devil (and Daniel
Webster)
when computing deflections and stresses for plates! I recommend
parabolic
tetrahedrons or hexahedrons in contrary for (thick) plate calculations,
that
means more input expense but the results are always save and free of
suspicious
interpretation constraints.