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 $\frac{N}{m}$
• Fixed temperature of 303.15 $K$ on all faces.

Reference Solution

The analytical solution for the unit stress is given by the following equation:

$$\sigma =\Delta T\times \gamma \times E$$

where:

• $\Delta T$: change in temperature = 10 $K$
• $\gamma$: thermal expansion coefficient = 1.2e-5 $\frac{1}{K}$
• $E$: Young’s modulus = 205 $GPa$

As a result, the analytical calculation gives a unit stress of: $$\sigma =24.6MPa$$

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 $[\mathrm{\%}]$
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