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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]\(^1\).

Geometry: Fixed Beam

A bar with a square cross-section was used for this case:

fixed beam used for the validation 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 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:

CaseMesh typeNumber of nodesElement type
A1st order tetrahedral14843D isoparametric
B2nd order tetrahedral93203D isoparametric
C1st order hexahedral3693D isoparametric
D2nd order hexahedral12213D isoparametric
Table 1: The details of each mesh that was simulated within the validation project
linear tetrahedral mesh created with the standard mesher
Figure 2: The 1st order mesh for case A, created with the Standard mesher algorithm in SimScale
linear hexahedral mesh created with Salome externally
Figure 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:

$$σ =ΔT \times \ γ \times \ E $$

where:

  • \(ΔT\): change in temperature = 10 \(K\)
  • \(γ\): thermal expansion coefficient = 1.2e-5 \(1 \over \ K\)
  • \(E\): Young’s modulus = 205 \(GPa\)

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
\([\%]\)
A24.624.60.00
B24.624.60.00
C24.624.60.00
D24.624.60.00
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.

Last updated: November 10th, 2020

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