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# 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:

## 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:

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

Last updated: July 21st, 2021