# Fixed Beam Under Changing Temperature¶

## Overview¶

The aim of this test case is to validate the following functions:

• thermomechanical analysis

The simulation results of SimScale were compared to the analytical results in [Roark]. The meshes used in (A) and (B) were created with the parametrized-tetrahedralization-tool on the SimScale platform and the meshes used in (C) and (D) were meshed with Salome.

Import validation project into workspace

## Geometry¶

Geometry of the beam

The bar has a cross section $$A$$ of 0.05 x 0.05 m² and a length $$l$$ = 1.0 m.

## Analysis type and Domain¶

Tool Type : Calculix, Code_Aster

Analysis Type : Thermomechanical

Mesh and Element types :

Case Mesh type Number of nodes Element type Tool type
(A) linear tetrahedral 84 3D isoparametric Calculix
(B) quadratic tetrahedral 369 3D isoparametric Calculix
(C) linear hexahedral 369 3D isoparametric Calculix
(D) quadratic hexahedral 1221 3D isoparametric Calculix
(E) linear tetrahedral 84 3D isoparametric Code_Aster
(F) quadratic tetrahedral 369 3D isoparametric Code_Aster
(G) linear hexahedral 369 3D isoparametric Code_Aster
(H) quadratic hexahedral 1221 3D isoparametric Code_Aster

Mesh used for the SimScale case (A)

Mesh used for the SimScale case (C)

## Simulation Setup¶

Material:

• isotropic: E = 205 GPa, $$\nu$$ = 0.3
• thermal expansion coefficient $$\gamma$$ = 12E-6 1/K

Constraints (Boundary Conditions):

• Fixed x-translation of face ABCD and face A’B’C’D’
• Additional points were fixed to restrain the remaining translations and rotations

• Temperature change $$\Delta T$$ = 10 K on the outer surface

## Reference Solution¶

$$$\label{ref1} \sigma = \Delta T \gamma E$$$

The equation $$\eqref{ref1}$$ used to solve the problem is derived in [Roark]. Inserting the values described in the previous chapter results in the unit stress $$\sigma$$ = 24.6 Mpa in the whole beam.

## Results¶

Comparison of the unit stress $$\sigma$$ in the beam obtained with SimScale with the results derived from the equations presented in [Roark].

Comparison of the unit stress $$\sigma$$ in [MPa]
Case [Roark] SimScale Error (%)
(A) 24.6 24.6 0.00%
(B) 24.6 24.6 0.00%
(C) 24.6 24.599 0.004%
(D) 24.6 24.599-24.601 $$\pm$$ 0.004%
(E) 24.6 24.6 0.00%
(F) 24.6 24.6 0.00%
(G) 24.6 24.6 0.00%
(H) 24.6 24.6 0.00%

Note: The worst results was obtained with mesh (D). This was due to an interpolation error and is shown in the picture below.

Stress filed obtained with mesh (D)

## References¶

 [Roark] (1, 2, 3, 4) (2011)”Roark’s Formulas For Stress And Strain, Eighth Edition”, W. C. Young, R. G. Budynas, A. M. Sadegh