This example concerns a torsion problem of an embedded plate subjected to two
concentrated loads. This example illustrates the role of the different shell element
formulations with regard to the mesh.
The boundary
conditions are such that the three nodes of a single side and the two
middle ones are blocked, while the others are free with respect to the Y
axis.
The material used follows a linear elastic behavior with the following characteristics:
Material Properties
Value
Initial density
7.8x10-3
Young's modulus
210000
Poisson ratio
0.3
Figure 2. Geometry of the Problem
Model Method
Four different types of mesh are used:
Mesh 1
Two quadrilateral shells and four triangular shells (2Q4-4T3)
Mesh 2
Four quadrilateral shells (4Q4)
Mesh 3
Eight triangular shells (8T3)
Mesh 4
Eight triangular shells (8T3 inverse)
For each model, the following shell formulations are tested:
QBAT formulation (Ishell
=12)
QEPH formulation (Ishell
=24)
Belytshcko & Tsay formulation (Ishell =1
or 3, hourglass control TYPE1, TYPE3)
C0 and DKT18 formulations
Figure 3. Square Plate Meshes
Results
Curves and Animations
This example compares several models concerning:
the use of different element formulations for each mesh
the different types of mesh for a given element formulation
To compare the results, two criteria are used:
absorbed energy (internal and hourglass)
vertical displacement of the node under the loading point
The following diagrams summarize the results obtained.
Energy Curves / Comparison for Element Formulations
Mesh 1: 2Q4-4T3Figure 4. Internal Energy for 2 x Q4 and 4 x T3 Elements
Mesh 2: 4Q4Figure 5. Internal Energy for 4 x Q4 Elements
Meshes 3 and 4: 8T3 and 8T3_INVFigure 6. Internal Energy for 8 x T3 Elements
Energy Curves / Comparison for Mesh Definitions
Figure 7. Internal Energy for Different Meshes Figure 8. Hourglass Energy for Different Meshes
Table 1. Displacement and Maximum Energy Comparison
2 Q4- 4
T3
4 Q4
8 T3
8 T3
Inverse
QEPH
BT_TYPE1
BT_TYPE4
BATOZ
QEPH
BT_TYPE1
BT_TYPE4
BATOZ
DKT
C0
DKT
C0
IEmax
2.74x10-2
2.35x10-2
2.37x10-2
7.21x10-2
3.64x10-2
2.93x10-2
2.97x10-2
2.30x10-2
1.37 x10-1
1.69x10-2
1.37x10-1
1.69x10-2
HEmax
--
1.01x10-4
1.03x10-4
--
--
1.94x10-4
1.98x10-6
--
--
--
--
--
DZmax
1.75x10-3
1.78x10-3
1.78x10-3
1.21x10-2
2.42x10-3
2.95x10-3
2.97x10-3
2.30x10-3
1.44x10-2
1.69x10-3
1.44x10-2
1.69x10-3
Conclusion
A square plate under torsion is a severe test to study the behavior of shell elements in
torsion-bending. A general overview of the results obtained highlight the following
key points:
For the 4Q4 mesh, the results obtained using QBATOZ and QEPH are similar. BT
elements are too flexible and are not significantly influenced by the
hourglass formulation, due to the in-plane mesh.
For triangular meshes, the DKT element is able to bend much better, the
co-element being too stiff.
The mesh with both Q4 and T3 elements may not comment like the other two, as
one part uses the triangle elements employed in Radioss.