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.

  • Concentrated loads (/CLOAD)
    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

    Figure 1. Boundary Conditions and Loads
  • Element formulation (Properties)

Input Files

The input files used in this example include:
8T3 inv

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 [ g m m 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada WcaaqaaiaadEgaaeaacaWGTbGaamyBamaaCaaaleqabaGaaG4maaaa aaaakiaawUfacaGLDbaaaaa@3BBC@
Young's modulus
210000 [ MPa ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaai Gac2eacaGGqbGaaiyyaaGaay5waiaaw2faaaaa@3BE6@
Poisson ratio

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


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: 2Q4-4T3

Figure 4. Internal Energy for 2 x Q4 and 4 x T3 Elements
Mesh 2: 4Q4

Figure 5. Internal Energy for 4 x Q4 Elements
Meshes 3 and 4: 8T3 and 8T3_INV

Figure 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
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


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.