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] $^{1}$ .

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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

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.

Case	Mesh Type	Number of Nodes	Element Type
(A)	Standard	32553	1st order tetrahedral
(B)	Standard	239421	2nd 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 $G P a$ , $v$ = 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 $M P a$ 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 $d x [m]$	5.90e-05	5.72e-05	-3.05
(B)	Displacement $d x [m]$	5.90e-05	5.72e-05	-3.05
(A)	Cauchy Stress $σ_{x x} [M P a]$	-6.00e01	-5.89e01	-1.83
(B)	Cauchy Stress $σ_{x x} [M P a]$	-6.00e01	-5.99e01	-0.16
(A)	Cauchy Stress $σ_{y y} [M P a]$	1.00e02	1.00e02	0
(B)	Cauchy Stress $σ_{y y} [M P a]$	1.00e02	1.00e02	0
(A)	Total Strain $ϵ_{x x}$	-4.50e-04	-4.63e-04	2.89
(B)	Total Strain $ϵ_{x x}$	-4.50e-04	-4.67e-04	3.78
(A)	Total Strain $ϵ_{y y}$	5.90e-04	5.72e-04	-3.05
(B)	Total Strain $ϵ_{y y}$	5.90e-04	5.72e-04	-3.05

Table 3: Comparison of results at point A

Find in Figure 3 below the case B results for the total strain $ϵ_{y y}$ distributed across the cylinder:

contours of strain simscale post processing — Figure 3: Contours of total strain $ϵ_{y y}$ acting on the hollow cylinder under a plane strain condition.

References

https://www.code-aster.org/V2/doc/v14/en/man_v/v3/v3.04.004.pdf

Last updated: July 21st, 2021

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