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Documentation

Validation Case: Cantilever Beam With Off-Center Masses – NAFEMS Test FV4

This validation case belongs to solid mechanics, with the case of a circular cantilever beam with two off-center point masses connected at the free end. The test case aims to validate the following parameters:

  • Point Mass boundary condition

The simulation results of SimScale were compared to the reference results presented in [NAFEMS]\(^1\)

Geometry

The geometry used for the case is as follows:

cantilever beam geometry
Figure 1: Geometry model of the cantilever beam

It consists of a straight beam of length 10 \(m\) and a circular, constant cross-section of radius 0.5 \(m\).

Analysis Type and Mesh

Tool Type: Code_aster

Analysis Type: Frequency Analysis

Mesh and Element Types:

The tetrahedral meshes were computed using SimScale’s standard mesh algorithm and manual sizing. The following table shows an overview of the mesh characteristics:

CaseMesh TypeNumber of NodesElement Type
AStandard32 6631st order tetrahedral
BStandard118 6961st order tetrahedral
CStandard240 7572nd order tetrahedral
Table 1: Number of mesh nodes and types of elements for each case.
cantilever beam mesh
Figure 2: Tetrahedral mesh for the beam, in this case corresponding to case C

Simulation Setup

Material:

  • Linar Elastic Isotropic
    • \(E\) = 200 \(GPa\)
    • \(\nu\) = 0.3
    • \(\rho\) = 8000 \(kg/m^3\)

Boundary Conditions:

  • Point Mass:
    • 1000 \(kg\) connected to face B at a distance of 2 \(m\)
    • 10000 \(kg\) connected to face B at a disance of 2 \(m\)
simscale validation two off-center point masses boundary conditions
Figure 3: Boundary conditions and remote point mass locations

Reference Solution

The reference solution presented in [NAFEMS]\(^1\) is of the numerical type. It is presented as the first six natural oscillation frequencies of the cantilever beam with the connected remote point mass. The values are presented in Table 2 alongside the comparison with SimScale results.

Results Comparison

Comparison of the first six natural frequencies for the run cases with the reference solution:

ModeNAFEMSCase A
[Deviation]
Case B
[Deviation]
Case C
[Deviation]
11.7231.739
[+0.9%]
1.731
[+0.5%]
1.767
[+2.6%]
21.7271.750
[+1.3%]
1.734
[+0.4%]
1.771
[+2.5%]
37.4137.573
[+2.1%]
7.448
[+0.5%]
7.593
[+2.4%]
49.97210.040
[+0.7%]
9.982
[+0.1%]
10.186
[+2.1%]
518.15518.344
[+1.0%]
18.141
[-0.1%]
18.524
[+2.0%]
626.95726.941
[-0.1%]
26.768
[-0.7%]
27.302
[+1.3%]
Table 2. Natural frequencies ( \(Hz\) ) results comparison

Following are the deformed shape plots for each mode, as taken from case C:

simscale validation remote mass mode shape 1
Figure 4: Mode shape 1, 1.767 \(Hz\) , simple bending in the XY plane
simscale validation off-center point mass mode shape 2
Figure 5: Mode shape 2, 1.771 \(Hz\) , simple bending in the XZ plane
simscale validation remote mass mode shape 3
Figure 6: Mode shape 3, 7.539 \(Hz\) , bending in the XZ plane, plus torsion
simscale validation off-center point mass mode shape 4
Figure 7: Mode shape 4, 10.186 \(Hz\) , double bending in the XY plane
simscale validation off-center point mass mode shape 5
Figure 8: Mode shape 5, 18.524 \(Hz\) , double bending in the XZ plane and torsion
simscale validation remote mass mode shape 6
Figure 9: Mode shape 6, 27.302 \(Hz\), double bending in the XY plane, plus torsion

References

  • NAFEMS publication TNSB, Rev. 3, “The Standard NAFEMS Benchmarks”, October 1990.

Last updated: December 6th, 2021

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