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Documentation

# Validation Case: Static Analysis of an I-Beam Under Remote Force

This validation case belongs to solid mechanics. This test case aims to validate the following parameters:

The simulation results of SimScale were compared to the results derived from [Roark]$$^1$$.

## Geometry

The geometry used for the case is as follows:

The beam has a length $$L$$ of 1 $$m$$, with the cross-section dimensions as shown:

## Analysis Type and Mesh

Tool Type: Code_Aster

Analysis Type: Static Linear

Mesh and Element Types:

Tetrahedral meshes were computed using SimScale Standard mesh algorithm and manual sizing. The table below shows an overview of the mesh characteristics. Figure 3: Second order tetrahedral mesh used in case B

## Simulation Setup

Material:

• Linear Elastic Isotropic:
• $$E =$$ 205 $$GPa$$
• $$\nu =$$ 0.28

Boundary Conditions:

• Constraints:
• Face A is fixed
• Remote force of 1000 $$N$$ applied at a distance $$d =$$ 1 $$m$$ on face B

## Reference Solution

The analytical solutions for the deflection $$w$$ at the free end of the beam are given by the following equations. The remote force $$F$$ is substituted with a force and moment pair $$M$$:

$${M = Fd} \tag{1}$$

$$w = \frac{F L^3}{3 E I} + \frac{M L^2}{2 E I} \tag{2}$$

$$I = \frac{1}{12} ( B H^3 – b h^3) \tag{3}$$

The computed reference solution is:

$$M = 1000\ Nm$$

$$I = 1.84×10^{-6}\ m^4$$

$$w = 2.209×10^{-3}\ m$$

## Result Comparison

Comparison of displacement DZ of the center point of face B with the computed reference solution $$w$$:

Comparison of the neutral fiber deflection shapes: Figure 5: Neutral fiber deflection comparison for analytical and computed solutions

And finally, a plot showing the deformed shape and the magnitude of DZ displacement: Figure 6: Case B, showing the deformed shape of the beam with contours for the displacement in the z-direction

References

• (2011) “Roark’s Formulas For Stress And Strain, Eighth Edition”, W. C. Young, R. G. Budynas, A. M. Sadegh

Note

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Last updated: September 9th, 2022