
                             SLFFEA 1.1 Shell README

                    Copyright (C) 1999 San Le

                    email:     slffea@yahoo.com
                    homepage:  http://www.geocities.com/Athens/2099/slffea.html


Of all the elements, this is the one I am most unsure with in terms of accuracy.
While the results compare well to the benchmark plate problem, when I try doing
the roof problem as given in:

Zienkiewicz, O. C., R. L. Taylor, and J. M. Too, Analysis of Thick and Thin Shell
        Structures by Curved Finite Elements, International Journal for Numerical
        Methods in Engineering, 3 (1971), 275-290.

my results are somewhat unsatisfactory.  The problem involves a roof deforming
under its own weight.  Because the weight is given per unit
surface area, a best guess had to be made for how it would translate into nodal
forces.  The results I got were that the element's vertical displacement on center
nodes, although qualitatively correct, are about half as much as Zienkiewicz's
results when 16 elements define the mesh.   Even when the number of elements was
increased to 64, there was no significant difference in the calculated deflection
of the center.  The cause of the inaccuracy may be from an incorrect reduction of
the distributed weight to equivalent nodal forces, or that the element is exhibiting
overly stiff behavior.  It is important to note that Zienkiewicz used parabolic and
cubic elements which will probably give better results. 

Shell elements are definitely the hardest to implement, even harder than the
non-linear brick.  I have also heard that there has been changes in the theory
from the 1987 version given by Hughes in [12] which I have used.  So you may
want to do some benchmark test for yourself to determine whether it meets
your standards of accuracy.  My main issue with the theory as given in Hughes
is that the local axes are treated as constants.  For instance, in equation
6.2.45 on page 393, the partial derivatives are taken only over the shape 
functions Na and za, but not q and e(the local axes of the lamina and fiber
respectively) which are treated as constants.  This may have been done to
simplify the resulting equations as well as maintain the 2X2 Gauss integration
rule, but it does seem incorrect that these local axes aren't being interpolated
between nodes like other quantities.  If there is someone who wants to discuss
this more, just email me.  I'm currently too lazy to implement a more correct
version of the element by making the local axes a function of x,y,z.

Also you should note that stresses and strains are given relative to the
local lamina coordinate system.  The significance of this comes across
in the input file "quad" which is the same as the "qdinc" problem for the
quadrilateral elements.  Note that Stress and Strain YY for this shell element
seem to be switched with that of Stress and Strain XX, in terms of the solution
using quadrilateral elements.  I suspect this is due to a switch of axes
which occurs in the lamina coordinate system when the calculations are made.
After a careful examination of the rotation matrices, I see that for element
66, the rotation matrices for bending in the lamina(rotate.l_bend.) look like:

  0.046447   0.998921   0.000000  
 -0.998921   0.046447  -0.000000  
 -0.000000   0.000000   1.000000  
 
  0.046530   0.998917  -0.000000  
 -0.998917   0.046530   0.000000  
  0.000000  -0.000000   1.000000  
 
  0.084252   0.996444  -0.000000  
 -0.996444   0.084252  -0.000000  
 -0.000000   0.000000   1.000000  
 
  0.084169   0.996451  -0.000000  
 -0.996451   0.084169  -0.000000  
 -0.000000   0.000000   1.000000  
 
  0.046447   0.998921   0.000000  
 -0.998921   0.046447  -0.000000  
 -0.000000   0.000000   1.000000 

for the first 4 integration points(The second four are identical).  Since each
row represents a lamina basis vector, it is clear that the x and y lamina basis
vectors are almost exactly perpendicular with that of the global basis given by:

  1.000000   0.000000   0.000000  
  0.000000   1.000000   0.000000  
  0.000000   0.000000   1.000000 

For instance, for the first integration point, lamina basis x vector is given by:

  0.046447   0.998921   0.000000  

showing that it points in positive global y, while lamina basis y vector

 -0.998921   0.046447  -0.000000  

points in negative global x.  This is expected.  As you follow the lamina axes
beginning with the first element, which are almost identical with the global
axes, and you proceed to element 66, you will notice that the lamina axes x
and y are gradually rotating 90 degrees counter-clockwise, until lamina x points
in global y and lamina y points in negative global x.  Finally, another source of
discrepancy may come from the effect of under-integrating in the shell.

Except for the above, further proof that the results are consistent is the fact
that the displacements and the principal stresses are the same(principal strains
are also very similar). 

Again, the patch test is divided into one for angles and one for displacement.  The
patch test I did was the same as that of the plate except that the mesh is out of 
the x-y plane, but all 4 elements still share the same plane.  My shell element passed
this test.  At first, I tried to do the test with each of the 4 elements in a
different plane.  I do not think this is a valid test though, since depending on
how the element is oriented, the load on it will be either supported by shear or
bending, so you get results inconsistent with the patch test predictions.  This is
what I experienced.

It is interesting to note that when you calculate stresses based on node points
rather than gauss points, the shear terms for the roof problem are qualitatively
correct for XX, YY, XY, I, and II but for ZX and YZ, they are very different.  Node
stress and strain ZX and YZ are orders of magnitude larger than when 
under integration is used.  This is expected, since as you know, the purpose
of under integrating is to prevent shear locking.

Nodal averaging is done for this element even though stresses and strains are highly
localized.  I feel this is justified because the local coordinate systems
should ideally be continuous across element boundaries, and therefore their local
stresses and strains should be continuous as well.  If someone has more insight into
this, please email me to discuss it.

