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:

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

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 \([\%]\)
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

As shown in Table 2, the SimScale results perfectly match the analytical ones with 0.00% error.

Last updated: February 5th, 2021