Fixed Beam Under Changing Temperature

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

The bar has a cross section A

$A$

 of 0.05 x 0.05 m² and a length l

$l$

 = 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

Simulation Setup

Material:

• isotropic: E = 205 GPa, ν
  $\nu$

   

  = 0.3

• thermal expansion coefficient γ
  $\gamma$

   

  = 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

Loads:

• Temperature change ΔT
  $\mathrm{\Delta }T$

   

  = 10 K on the outer surface

Reference Solution

σ=ΔTγE(1)

$$\begin{array}{}\text{(1)}& \sigma =\mathrm{\Delta }T\gamma E\end{array}$$

 

The equation (1)

$\text{(1)}$

 used to solve the problem is derived in [Roark]. Inserting the values described in the previous chapter results in the unit stress σ

$\sigma$

 = 24.6 Mpa in the whole beam.

Results

Comparison of the unit stress σ

$\sigma$

 in the beam obtained with SimScale with the results derived from the equations presented in [Roark].

Comparison of the unit stress σ

$\sigma$

 

 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.

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