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Validation Case: Static Analysis of an I Beam under Remote Force

This validation case belongs to solid mechanics. The aim of this test case is to validate the following parameters:

  • Remote force boundary condition

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 under remote force validation case
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. Hexahedral meshes were computed locally and uploaded into the simulation project.

CaseMesh TypeNumber of
Nodes
Element Type
A1st Order Tetrahedral 3345Standard
B1st Order Hexahedral5757Standard
C2nd Order Tetrahedral20742Standard
D2nd Order Hexahedral20749Standard
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 on case C.
hexahedral mesh static analysis of i beam under remote force validation case
Figure 4: Second order hexahedral mesh used on case D.

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 under remote force validation case
Figure 5: Remote force diagram showing the applied load on the beam.

Reference Solution

The analytical solutions for the deflection \(w\) 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 %
B2.199242.2090.44 %
C2.21212.209-0.14 %
D2.212342.209-0.15 %
Table 3: Deflection results comparison

Comparison of the neutral fiber deflection shapes:

neutral fiber deflection static analysis of i beam under remote force validation case
Figure 6: 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 7: Deformed shape plot with contour for DZ displacement.

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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