The 3D geometry is a 45\(^0\) 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 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.
Case
Mesh Type
Number of Nodes
Element Type
(A)
linear tetrahedral
32553
Standard
(B)
quadratic tetrahedral
239421
Standard
(C)
linear hexahedral
768
Standard
(D)
quadratic hexahedral
2720
Standard
Table 2: Mesh characteristics.
Figure 2: Linear standard mesh used for case (A).
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].
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
(C)
Displacement \(dx\ [m]\)
5.90e-05
5.71e-05
-3.22
(D)
Displacement \(dx\ [m]\)
5.90e-05
5.71e-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 \(\sigma_{yy}\ [MPa]\)
1.00e02
1.00e02
0
(B)
Cauchy Stress \(\sigma_{yy}\ [MPa]\)
1.00e02
1.00e02
0
(C)
Cauchy Stress \(\sigma_{yy}\ [MPa]\)
1.00e02
1.03e02
3
(D)
Cauchy Stress \(\sigma_{yy}\ [MPa]\)
1.00e02
0.99e02
-1
(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
(C)
Total Strain \(\epsilon_{xx} \)
-4.50e-04
-4.34e-04
-3.55
(D)
Total Strain \(\epsilon_{xx} \)
-4.50e-04
-4.67e-04
3.78
(A)
Total Strain \(\epsilon_{yy} \)
5.90e-04
5.72e-04
-3.05
(B)
Total Strain \(\epsilon_{yy} \)
5.90e-04
5.72e-04
-3.05
(C)
Total Strain \(\epsilon_{yy} \)
5.90e-04
5.71e-04
-3.22
(D)
Total Strain \(\epsilon_{yy} \)
5.90e-04
5.71e-04
-3.22
Table 3: Comparison of results at point A.
Find figure 4 below for the total strain \(\epsilon_{yy}\) distributed across the cylinder:
Figure 4: Contours of total strain \(\epsilon_{yy} \) acting on the hollow cylinder.
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