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. Details can be found in Table 1:

Case	Mesh type	Number of nodes	Element type
A	Standard	1484	1st order tetrahedral
B	Standard	9320	2nd order tetrahedral

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

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

Last updated: July 21st, 2021