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Validation Case: Hollow Cylinder in Plane Strain Condition

This plane strain condition validation case belongs to solid mechanics. This test case aims to validate the following parameters:

  • Distributed pressure
  • Symmetry boundary condition
  • Nodal displacement
  • Strains and Nodal stresses

The simulation results from SimScale were compared to the analytical results presented in [SSLV04]\(^1\).

Geometry

The geometry used for the case is as follows:

hollow cylinder geometry
Figure 1: Hollow cylinder geometry wireframe

The 3D geometry is a 45-degree section of a hollow cylinder with dimensions as tabulated below:

ABEFA’B’E’F’
x0.10.20.07070.14140.10.20.07070.1414
y000.07070.1414000.07070.1414
z00000.010.010.010.01
Table 1: Geometry dimensions in meters

Analysis Type and Mesh

Tool Type: Code Aster

Analysis Type: Static linear

Mesh and Element Types: The meshes used in this project were created using the standard meshing tool on the SimScale platform. In Table 2 you will find a summary of the mesh characteristics.

CaseMesh TypeNumber of NodesElement Type
(A)Standard325531st order tetrahedral
(B)Standard2394212nd order tetrahedral
Table 2: Mesh characteristics

Find below the manually refined 1st order standard mesh used in case A:

linear standard mesh in simscale for plane strain validation case
Figure 2: First order standard mesh used for case A

Simulation Setup

Material:

  • Steel (linear elastic)
    • \(E\) = 200 \(GPa\), \(v\) = 0.3

Boundary Conditions:

  • Constraints
    • Face EFE’F’ zero normal-displacement
    • Face ABA’B’ zero y-displacement
    • Face ABEF and face A’B’E’F’ zero z-displacements fixed
  • Loads
    • The pressure of 60 \(MPa\) on Face AEA’E’

Reference Solution

The analytical solution is given by the equations presented under Reference Solution\(^1\).

Result Comparison

The results obtained from SimScale for displacements, stresses, and strains at point A are compared with those presented in SSLV04.

CaseQuantity[SSLV04]SimScaleError [%]
(A)Displacement \(dx\ [m]\)5.90e-055.72e-05-3.05
(B)Displacement \(dx\ [m]\)5.90e-055.72e-05-3.05
(A)Cauchy Stress \(\sigma_{xx}\ [MPa]\)-6.00e01-5.89e01-1.83
(B)Cauchy Stress \(\sigma_{xx}\ [MPa]\)-6.00e01-5.99e01-0.16
(A)Cauchy Stress \(\sigma_{yy}\ [MPa]\)1.00e021.00e020
(B)Cauchy Stress \(\sigma_{yy}\ [MPa]\)1.00e021.00e020
(A)Total Strain \(\epsilon_{xx} \)-4.50e-04-4.63e-042.89
(B)Total Strain \(\epsilon_{xx} \)-4.50e-04-4.67e-043.78
(A)Total Strain \(\epsilon_{yy} \)5.90e-045.72e-04-3.05
(B)Total Strain \(\epsilon_{yy} \)5.90e-045.72e-04-3.05
Table 3: Comparison of results at point A

Find in Figure 3 below the case B results for the total strain \(\epsilon_{yy}\) distributed across the cylinder:

contours of strain simscale post processing
Figure 3: Contours of total strain \(\epsilon_{yy} \) acting on the hollow cylinder under a plane strain condition.

Last updated: July 21st, 2021

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