VM198

VM198
Large Strain In-plane Torsion Test

Overview

Reference:J. C. Nagtegaal, J. E. DeJong, "Some Computational Aspects of Elastic-Plastic Strain Analysis", Intl J. of Numerical Methods in Engineering, Vol. 17, 1981, pp. 15–41.
Analysis Type(s):Static Analysis (ANTYPE = 0)
Element Type(s):
2D 4-Node Structural Solid Elements (PLANE182)
2D 8-Node Structural Solid Elements (PLANE183)
3D 8-Node Structural Solid Elements (SOLID185)
3D 20-Node Structural Solid Elements (SOLID186)
Input Listing:vm198.dat

Test Case

A hollow, thick-walled, long cylinder made of an elastoplastic material is under an in-plane torsional loading which causes the inner surface of the cylinder to undergo a rotation of 60°. Find the maximum shear stress (τmax) developed at the inner surface at the end of loading.

Figure 304: Large Strain In-plane Problem Sketch

Large Strain In-plane Problem Sketch

Material PropertiesGeometric PropertiesLoading
E = 7200 psi
μ = 0.33
σyp = 10 psi
ET = 40 psi
R1 = 10 in
R2 = 20 in
Θ = 60°
At nodes on the inner surface in 10 equal load steps along the circumference

Analysis Assumptions and Modeling Notes

The problem is solved for the following cases:

Case 1: Using PLANE182 elements with plane strain element behavior
Case 2: Using SOLID185 elements
Case 3: Using PLANE183 elements with plane strain element behavior
Case 4: Using PLANE182 elements with torsion
Case 5: Using SOLID185 elements extruded from PLANE182 element solution with torsion using MAP2DTO3D
Case 6: Using PLANE183 elements with torsion
Case 7: Using SOLID186 elements extruded from PLANE183 element solution with torsion using MAP2DTO3D

The plasticity is modeled using the bilinear isotropic hardening rule. The plane strain condition is assumed for cases 1 and 3 along the length of the cylinder. Due to the axisymmetric loading, only a small portion (3° span) of the cross-section is modeled for cases 1, 2, and 3 using ten elements. Nodal rotations and displacement couplings for cases 1, 2, and 3 are employed to ensure the circumferential symmetry in the deformed configuration. In addition, a character parameter for degrees of freedom is used in the CP commands and in the macro SOLD for cases 1, 2, and 3.

Since the modeling and loading is axisymmetric, the global X-direction represents the radial direction, the global Y-direction represents the axial direction, and the negative of the global Z-direction normal and into the 2D X-Y plane rotated about the global Y-axis represents the circumferential or hoop direction.

For cases 4 and 6, 40 elements along the radial direction are used to model the cut section on the 2D X-Y plane with only one layer of elements required along the axial direction. The ROTY rotation degrees of freedom available with PLANE182 and PLANE183 axisymmetric elements with torsion are used to model the high 60° twist of the inner radial surface into the X-Y plane (or hoop direction). MAP2DTO3D is then used to map the results from the 2D model to the 3D model for cases 5 and 7.

The maximum shear stress is calculated from the solution results for all cases. To maintain consistency with the reference solution, the maximum shear stress in a negative direction is observed.

Results Comparison

 TargetMechanical APDLRatio
PLANE182Shear Stressmax , psi-48.0-46.50.969
PLANE183Shear Stressmax , psi-48.0-45.90.956
SOLID185Shear Stressmax , psi-48.0-46.30.964
PLANE182 with torsionShear Stressmax , psi-48.0-46.40.967
PLANE183 with torsionShear Stressmax , psi-48.0-47.40.987
SOLID185 extruded from PLANE182 with torsion using MAP2DTO3DShear Stressmax , psi-48.0-47.00.979
SOLID186 extruded from PLANE183 with torsion using MAP2DTO3DShear Stressmax , psi-48.0-47.40.987

Figure 305: Typical Element Deformation Display Using PLANE182 Elements

Typical Element Deformation Display Using PLANE182 Elements

Figure 306: Typical Stress vs. Rotation Display Using PLANE182 Elements

Typical Stress vs. Rotation Display Using PLANE182 Elements