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:

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

	A	B	E	F	A’	B’	E’	F’
x	0.1	0.2	0.0707	0.1414	0.1	0.2	0.0707	0.1414
y	0	0	0.0707	0.1414	0	0	0.0707	0.1414
z	0	0	0	0	0.01	0.01	0.01	0.01

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.

Case	Mesh Type	Number of Cells	Number of Nodes	Element Type
(A)	Standard	150813	30834	1st order tetrahedral
(B)	Standard	150813	222478	2nd order tetrahedral

Find below the manually refined 2st order standard mesh used in Case B:

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.

Case	Quantity	[SSLV04]	SimScale	Error [%]
(A)	Displacement \(dx\ [m]\)	5.90e-05	5.72e-05	-3.05
(B)	Displacement \(dx\ [m]\)	5.90e-05	5.72e-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 \(\sigma_{yy}\ [MPa]\)	1.00e02	1.00e02	0
(B)	Cauchy Stress \(\sigma_{yy}\ [MPa]\)	1.00e02	1.00e02	0
(A)	Total Strain \(\epsilon_{xx} \)	-4.50e-04	-4.63e-04	2.89
(B)	Total Strain \(\epsilon_{xx} \)	-4.50e-04	-4.67e-04	3.78
(A)	Total Strain \(\epsilon_{yy} \)	5.90e-04	5.75e-04	-2.54
(B)	Total Strain \(\epsilon_{yy} \)	5.90e-04	5.72e-04	-3.05

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

Last updated: June 15th, 2025