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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:

geometry model static analysis of i-beam
Figure 1: Geometry model of an I beam

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

cross section parameters static analysis of i beam under remote force validation case
Figure 2: Beam cross-section dimensions parameters
ParameterValue [\(m\)]
B0.06
H0.08
b0.04
h0.06
Table 1: Cross-section geometry parameters

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.

CaseMesh TypeNumber of
Nodes
Element Type
AStandard 33451st order tetrahedral
BStandard207422nd order tetrahedral
Table 2: Number of mesh nodes and types of elements for each case
tetrahedral mesh static analysis of i beam under remote force validation case
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
  • Loads:
    • Remote force of 1000 \(N\) applied at a distance \(d = \) 1 \(m\) on face B
remote force static analysis of i-beam validation
Figure 4: Remote force diagram showing the applied load on the beam

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 \(w\):

CASEDZ
[\(10^{-3}\ m\)]
\(w\)
[\(10^{-3}\ m\)]
Error
A2.141092.2093.07 %
B2.21212.209-0.14 %
Table 3: Deflection results comparison

Comparison of the neutral fiber deflection shapes:

neutral fiber deflection static analysis of i-beam
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:

deformed and colored shape plot static analysis of i beam under remote force validation case
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

If you still encounter problems validating your simulation, then please post the issue on our forum or contact us.

Last updated: July 21st, 2021

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