
This test is used to study the convergence and the absence
of any locking in the case of very thin shells (th / L = 0.5e^{4}).
You will use 2D meshes. 

Reference:
TIMOSHENKO S.P., WOINOWSKYKRIGER S., Theory of
Plates and Shells, 1959. 

Specifications
Geometry Specifications
Length:
L = 200 mm 

Thickness:
th = 0.01 mm 

For symmetry reasons, only a quarter of the plate is
modeled (ABCD). 
Analysis Specifications
Young Modulus (material):
E = 78000 MPa 
Poisson's Ratio (material):
ν=
0.3 
Mesh Specifications:
Imposed numbers of nodes (see the tables of results) 

Restraints (Userdefined):

Point A (position fixed): Tx=Ty=0

Edge AB (simply supported edge): Tz=Ry=0

Edge DA (simply supported edge): Tz=Rx=0

Edge BC (symmetry plane edge): Tx=Ry=Rz=0

Edge CD (symmetry plane edge): Ty=Rx=Rz=0

Loads:



Results

The analytical solutions for the z displacement of the
point C are:
Case 1: Tz=64.96 mm
Case 2: Tz=90.942 mm

The tables below present the analysis results for cases 1
and 2.
The normalized results (computed results divided by analytical solution)
are listed.
Nodes 
Analytical solution
[mm] 
Values 
Linear triangle
(TR3) 
Parabolic triangle
(TR6) 
Linear quadrangle
(QD4) 
Parabolic quadrangle
(QD8) 




Computed results
[mm] 
Normalized results 
Computed results
[mm] 
Normalized results 
Computed results
[mm] 
Normalized results 
Computed results
[mm] 
Normalized results 
3 x 3 
64.96 
62.15 
0.957 
62.09 
0.956 
67.44 
1.038 
63.03 
0.970 
5 x 5 
64.96 
64.66 
0.995 
64.61 
0.995 
66.37 
1.022 
64.58 
0.994 
7 x 7 
64.96 
64.81 
0.998 
64.83 
0.998 
65.77 
1.013 
64.80 
0.997 
9 x 9 
64.96 
64.87 
0.999 
64.89 
0.999 
65.49 
1.008 
64.87 
0.999 
11 x 11 
64.96 
64.91 
0.999 
64.92 
0.999 
65.33 
1.006 
64.91 
0.999 
Nodes 
Analytical solution
[mm] 
Values 
Linear triangle
(TR3) 
Parabolic triangle
(TR6) 
Linear quadrangle
(QD4) 
Parabolic quadrangle
(QD8) 




Computed results
[mm] 
Normalized results 
Computed results
[mm] 
Normalized results 
Computed results
[mm] 
Normalized results 
Computed results
[mm] 
Normalized results 
3 x 3 
90.942 
83.621 
0.919 
87.039 
0.957 
90.308 
0.993 
86.767 
0.987 
5 x 5 
90.942 
89.196 
0.981 
91.117 
1.002 
91.576 
1.007 
90.977 
1.000 
7 x 7 
90.942 
90.276 
0.993 
91.053 
1.001 
91.366 
1.005 
90.998 
1.001 
9 x 9 
90.942 
90.596 
0.996 
91.030 
1.001 
91.231 
1.003 
90.998 
1.001 
11 x 11 
90.942 
90.746 
0.998 
91.019 
1.001 
91.155 
1.002 
90.998 
1.001 


To Perform the Test:
The
simplysupportedplate11nodesqd8.CATAnalysis document presents a complete
analysis of this case, computed with a mesh formed of 11 nodes and
parabolic quadrangle elements (QD8).
To compute the case with other types of elements and other
number of nodes, proceed as follow:

Open the CATAnalysis document.

In the Advanced Meshing Tools workbench,
change the mesh specifications as indicated above.

Compute the case in the Generative Structural
Analysis workbench.
