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Documentation

Validation Case: Hollow Cylinder in Plane Strain Condition

This plane strain condition validation case belongs to solid mechanics. The aim of this test case is to validate the following parameters:

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

The simulation results of 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\(^0\) 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 case (A) and (B) were created using the standard mesher on the SimScale platform, while the meshes used in case (C) and (D) were created in an external platform and then imported to the SimScale workbench.

CaseMesh TypeNumber of NodesElement Type
(A)linear tetrahedral32553Standard
(B)quadratic tetrahedral239421Standard
(C)linear hexahedral768Standard
(D)quadratic hexahedral2720Standard
Table 2: Mesh characteristics.
linear standard mesh in simscale for plane strain validation case
Figure 2: Linear standard mesh used for case (A).
quadratic hexahedral mesh used for plane strain validation case in simscale
Figure 3: Quadratic hexahedral structured mesh used for case (D) imported to the SimScale workbench.

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
(C)Displacement \(dx\ [m]\)5.90e-055.71e-05-3.22
(D)Displacement \(dx\ [m]\)5.90e-055.71e-05-3.22
(A)Cauchy Stress \(\sigma_{xx}\ [MPa]\)-6.00e01-5.89e01-1.83
(B)Cauchy Stress \(\sigma_{xx}\ [MPa]\)-6.00e01-5.99e01-0.16
(C)Cauchy Stress \(\sigma_{xx}\ [MPa]\)-6.00e01-5.12e01-14.67
(D)Cauchy Stress \(\sigma_{xx}\ [MPa]\)-6.00e01-5.98e01-0.33
(A)Cauchy Stress \(\sigma_{yy}\ [MPa]\)1.00e021.00e020
(B)Cauchy Stress \(\sigma_{yy}\ [MPa]\)1.00e021.00e020
(C)Cauchy Stress \(\sigma_{yy}\ [MPa]\)1.00e021.03e023
(D)Cauchy Stress \(\sigma_{yy}\ [MPa]\)1.00e020.99e02-1
(A)Total Strain \(\epsilon_{xx} \)-4.50e-04-4.63e-042.89
(B)Total Strain \(\epsilon_{xx} \)-4.50e-04-4.67e-043.78
(C)Total Strain \(\epsilon_{xx} \)-4.50e-04-4.34e-04-3.55
(D)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
(C)Total Strain \(\epsilon_{yy} \)5.90e-045.71e-04-3.22
(D)Total Strain \(\epsilon_{yy} \)5.90e-045.71e-04-3.22
Table 3: Comparison of results at point A.
Figure 4: Contours of total strain \(\epsilon_{yy} \) acting on the hollow cylinder.

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