RD-E: 1802 Square Membrane Elastic

This example concerns the in-plane traction-comparison problem of an embedded plate subjected to two concentrated loads, as shown in Figure 2.

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, whereas the others are free with respect to the Y axis.

Two concentrated loads are applied on the corner points on opposing sides. They increase over time, as defined by the following function:
 F(t) 0 10 10 t 0 200 400

Input Files

Refer to Access the Model Files to download the required model file(s).

Model Description

Units: mm, ms, g, N, MPa

The material used follows a linear elastic behavior and has 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.

Mesh 1: 2Q4-4T3
Mesh 2: 4Q4

Energy Curves / Comparison for Mesh Definitions

Table 1. Displacement and Maximum Energy Comparison
Elastic Plate 2Q4-4T3 4Q4 8T3 8T3_INV
QEPH BT_TYPE 1 and 3 BATOZ QEPH BT_TYPE 1 and 3 BATOZ DKT CO DKT CO
IEmax 1.07 x 10-2 1.19 x 10-2 1.07 x 10-2 1.24 x 10-2 1.44 x 10-2 1.24 x 10-2 6.42 x 10-3 6.42 x 10-3 6.42 x 10-3 6.42 x 10-3
HEmax --- 2.10 x 10-5 -- -- 3.49 x 10-6 -- -- -- -- --
Dymax 1.18 x 10-3 (Traction) 1.38 x 10-3 (Traction) 1.18 x 10-3 (Traction) 1.24 x 10-3 1.44 x 10-3 1.24 x 10-3 6.42 x 10-3 6.42 x 10-3 6.42 x 10-3 6.42 x 10-3

Conclusion

In the case of elastic flat plate modeling, when the loading is in-plane, the shell elements are reduced to become a membrane if the loads applied do not cause buckling.

A general overview of the results obtained highlight the following key points:
• The quadrilateral shell elements QEPH and QBAT have the same in-plane behavior.
• The different types of hourglass formulations in the BT shell elements lead to the same results, as there is no out-of-plane deformation and the material is supposed to be elastic.
• The three in-plane behaviors of the DKT18 and T3C0 Radioss triangles are exactly the same, as both of the elements are used for the same membrane formulation.
• The triangles are stiffer than the quadrilateral elements and do not provide good results, especially when the mesh is coarse.

Refer to the Radioss Theory Manual for more details.