# RD-E: 1801 Square Plate Torsion

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.

## Options and Keywords Used

• Q4 shells
• T3 shells
• Hourglass and mesh
• Boundary conditions (/BCS)

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.

Two concentrated loads are applied on the corner points of the opposite side. They increase over time as defined by the following function:
 F(t) 0 10 10 t 0 200 400
• Element formulation (Properties)

## Input Files

Before you begin, copy the file(s) used in this example to your working directory.

## Model Description

Units: mm, ms, g, N, MPa

The material used follows a linear elastic behavior with the following characteristics:
Material Properties
Initial density
7.8x10-3 $\left[\frac{g}{m{m}^{3}}\right]$
Young's modulus
210000 $\left[\mathrm{MPa}\right]$
Poisson ratio
0.3

### Model Method

Four different types of mesh are used:
Mesh 1
Two quadrilateral shells and four triangular shells (2Q4-4T3)
Mesh 2
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

## 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)

The following diagrams summarize the results obtained.

### Energy Curves / Comparison for Element Formulations

Mesh 1: 2Q4-4T3
Mesh 2: 4Q4
Meshes 3 and 4: 8T3 and 8T3_INV

### Energy Curves / Comparison for Mesh Definitions

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.