Thermal Expansion of a Beam

This test lets you check analysis results for the thermal expansion of a beam, in the context of a static case. You will use 3D mesh with tetrahedron and hexahedron elements.

Reference:

Thermal Expansion Theory

 

Specifications

Geometry Specifications

Length:
L = 1000 mm

  • Model 3D

  • Model 2D

  • Model 1D

Section (square):
a = 100 mm


Analysis Specifications

Young Modulus (material):
E = 200 GPa



Model 3D - Case 1

Poisson's Ratio (material):
ν = 0.266

Thermal Expansion Coefficient:
α= 11.7 x 10-6/°

Mesh Specifications:
Size = 50 mm

Restraints:

  • 3D Model

    • Case 1: Ty = 0 on End 1, Tz = 0 on Face 1 and Tx = 0 on Face 2

    • Case 2: Ty  = 0 on End 1 and End 2, Tz = 0 on Face 1 and Tx = 0 on Face 2

    • Case 3: Ty  = 0 on End 1, Tz = 0 on Face 1 and Tx = 0 on Face 2

  • 2D Model

    • Case 1: Ty = 0 on Edge A, Tx = 0 on Edge C and Tz = 0 on the surface

    • Case 2: Ty  = 0 on Edge A and Edge B, Tx = 0 on Edge C and Tz = 0 on the surface

    • Case 3: Ty  = 0 on Edge A, Tx = 0 on Edge C and Tz = 0 on the surface

  • 1D Model

    • Case 1: Tz = 0 on A

    • Case 2: Ty  = 0 on A and B

    • Case 3: Ty  = 0 on A

Loads:

  • Case 1: Temperature Field T1 = 325 K

  • Case 2: Temperature Field T1 = 325 K

  • Case 3: Temperature Field T1 = 325 K
    and a Surface force density Fy = -5.85e+007 N_m2

The environment temperature is T0 = 300 K

 

Results

  • The analytic solution is:

Case 1:

Case 2:

Case 3: 

  • The computed results correspond to:
    • Uth: Maximal elongation at End 2 under thermal expansion effect.
    • σ22: Mean stress tensor of the beam, when the two extremities are clamped.
    • Uth- UFy: Maximal final displacement at End 2, when the force Fy is opposed to the thermal expansion.

The computed results are the same for the following elements: TE4, TE10, HE8, HE20, TR3, TR6, QD4, QD8, bar (linear and parabolic):

Case

Quantity

Analytical solution

Computed results

Case 1

Uth[mm]

0.2925

0.2925

Case 2

σ22 [MPa]

-58.5

-58.5

Case 3

Uth- UFy [mm]

0

0

To Perform the Test:

The Thermal_expansion_of_a_beamTE10.CATAnalysis document presents the complete analysis of this case, computed with a parabolic tetrahedron elements (TE10).

The Thermal_expansion_of_a_beamHE20.CATAnalysis document presents the complete analysis of this case, computed with a mesh formed of parabolic hexahedron elements (HE20).

To compute the case with linear elements, proceed as follow:

  1. Open one of the CATAnalysis documents.

  2. Enter the Advanced Meshing Tools workbench.

  3. In the specification tree, double-click the mesh:

    • Volume mesh  for linear tetrahedron elements (TE4).

    • Surface mesh for linear hexahedron elements (HE8).

    The Global Parameters dialog box appears.

  4. Select the Linear element type.

  5. Compute the case in the Generative Structural Analysis workbench.