Overview

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Physical Problem Description

This example concerns the torsion problem of an embedded plate subjected to two concentrated loads, as shown in the following diagram. This example illustrates the role of different shell element formulations with regard to the mesh.

Units:   mm, ms, g, N, MPa

The material used follows an isotropic elasto-plastic behavior with the Johnson-Cook plasticity model (/MAT/LAW2), with the following characteristics:

Fig 17: Geometry of the problem.

Analysis, Assumptions and Modeling Description

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Modeling Methodology

Four shells are modeled with different types of mesh:

For each model, the following element formulation is tested:

Fig 18: Square plate meshes.

RADIOSS Options Used

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:

 

Fig 19: Boundary conditions and loads.

Simulation Results

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Curves and Animations

This example compares several models concerning:

Two criteria used to compare the results are:

The following diagrams summarize the results obtained.

Energy Curves / Comparison for Element Formulations

Mesh 1:  2Q4-4T3

Fig 20: Internal energy for 2 x Q4 and 4 x T3 elements.

Fig 21: Hourglass energy for 2 x Q4 and 4 x T3 elements.

Fig 22: Force for 4 x Q4 elements.

Mesh 2:  4Q4

Fig 23: Internal energy for 4 x Q4 elements.

Fig 24: Hourglass energy for 4 x Q4 elements.

Fig 25: Total energy for 4 x Q4 elements.

Fig 26: Force for 4 x Q4 elements.

Fig 27: Internal energy for 8 x T3 elements.

Fig 28: Force for 8 x T3 elements.

Mesh 4:  8T3_INV

Fig 29: Internal energy for 8 x T3 elements (inversed mesh).

Fig 30: Force for 8 x T3 elements (inversed mesh).

Comparison for Different Meshes

Fig 31: Internal energy for different meshes.

Fig 32: Hourglass energy for different meshes.

Fig 33: Force for different meshes.

Displacement and Maximum Energy Comparison

Conclusion

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The purpose of this example was to highlight the role of the elasto-plastic treatment when formulating RADIOSS shells.  The in-plane plasticity was considered here.  Regarding the applied boundary conditions and the Poisson effect on the plate, the test may be very severe with respect to the behavior of plastic in under-integrated elements.

In the case of a mesh with four quadrilaterals, the QBAT element always provides the best results as it allows four integration points to be put over the element.  The plasticity computation over the integration points is thus more accurate. The under-integrated elements, having just one integration point at the center, allows only two integration points to be put through the width of the mesh. Another point concerns the role of Poisson’s ratio in the plasticity computation.  In fact, the QEPH element uses an analytical expression of the hourglass energy which takes into account the accurate expression in terms of the Poisson ratio (refer to the RADIOSS Theory Manual for further information).  However, some approximations are induced in its elasto-plastic formulation, possibly influencing the results, especially for low levels of work-hardening.  In the BT element formulation with a type 3 hourglass control, the Poisson ratio effect on the plastic part of the hourglass deformation is computed by a simplified expression which minimizes its role.  In fact, the results obtained using BT_TYPE3 are slightly affected by the change in (use =0 for the example studied and compare the results obtained).  The BT elements are generally more flexible and provide better results for a very coarse mesh.

For triangular meshes, the in-plane behavior of DKT18 should be noted as being the same as the T3C0 element. In fact, the elements are essentially different with respect to their bending behaviors.

When combining the T3 and Q4 elements, the results generally come between a uniform triangular mesh and a quadrangular mesh.