Validation Case: Fixed Beam Under Changing Temperature
This validation case belongs to solid mechanics. The aim of this test case is to validate the following parameters on a fixed beam, that is subjected to a temperature change of 10 \(K\):
Unit stress \((\sigma)\)
The simulation results of SimScale were compared to the results presented in [Roark]\(^1\).
A bar with a square cross-section was used for this case:
Figure 1: The fixed rectangular beam geometry with square cross-section
The cross section area of the bar is \(A\) = 0.05 x 0.05 \(m^2\) and it’s length is \(l\) = 1.0 \(m\).
Analysis Type and Mesh
Tool Type: Code_Aster
Analysis Type: Thermomechanical, linear with static inertia effect
Mesh and Element Types:
The meshes used in (A) and (B) were created with the Standard mesher tool on the SimScale platform and the meshes used in (C) and (D) were meshed locally and exported to SimScale. Details can be found in Table 1:
Case
Mesh type
Number of nodes
Element type
A
1st order tetrahedral
1484
3D isoparametric
B
2nd order tetrahedral
9320
3D isoparametric
C
1st order hexahedral
369
3D isoparametric
D
2nd order hexahedral
1221
3D isoparametric
Table 1: The details of each mesh that was simulated within the validation project
Figure 2: The 1st order mesh for case A, created with the Standard mesher algorithm in SimScaleFigure 3: The structured 1st order hexahedral element mesh used for case C, created externally
Simulation Setup
Material:
Steel (linear elastic)
isotropic: \(E\) = 205 \(GPa\), \(\nu\) = 0.3 and reference temperature \(T_o\) = 293.15 \(K\)
Boundary Conditions:
Fixed x-translation of face ABCD and face A’B’C’D’
Elastic support on the faces ABCD and A’B’C’D’, with a total isotropic stiffness of \(K\) = 1000 \(N \over \ m\)
Fixed temperature of 303.15 \(K\) on all faces.
Reference Solution
The analytical solution for the unit stress is given by the following equation:
As a result, the analytical calculation gives a unit stress of: $$σ =24.6\ MPa $$
Result Comparison
Comparison of the unit stress \(\sigma\) obtained from SimScale against the reference results obtained from [Roark]\(^1\) is given in the following table:
Case
[Roark] \([MPa]\)
SimScale \([MPa]\)
Error \([\%]\)
A
24.6
24.6
0.00
B
24.6
24.6
0.00
C
24.6
24.6
0.00
D
24.6
24.6
0.00
Table 2: Comparison of unit stress \(\sigma\) between SimScale simulation and [Roark]\(^1\)
As shown in Table 2, the SimScale results perfectly match the analytical ones with 0.00% error.
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