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Validation Case: Straight Beam with Damping of Rayleigh

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

  • Dynamic analysis
  • Material with Rayleigh damping model

The simulation results of SimScale were compared to the results presented in [SDLS123]\(^1\).

Geometry

The geometry used for the case is as follows:

geometrical model of beam with damping of rayleigh validation case
Figure 1: Geometrical model of the beam.

The beam has a length of 10 \(m\) and a square cross-section with a side length of 1 \(m\). The coordinates of each point can be found in the following table:

POINTX \([m]\)Y \([m]\)Z \([m]\)
A000
B1000
C1010
D010
E001
F1001
G1011
H011
Table 1: Geometry points coordinates

Analysis Type and Mesh

Tool Type: Code Aster

Analysis Type: Dynamic

Mesh and Element Types:

Tetrahedral meshes were computed using SimScale’s Standard mesh algorithm and manual sizing. Hexahedral meshes were locally computed and imported into the project.

CaseMesh TypeNumber
of Nodes
Element
Type
Integration
Scheme
A1st order Hexahedral496StandardExplicit
B2nd order Hexahedral1720StandardExplicit
C1st order Tetrahedral2862StandardExplicit
D2nd order Tetrahedral1575StandardExplicit
E2nd order Hexahedral1720StandardImplicit
F2nd order Tetrahedral1575StandardImplicit
Table 2: Mesh details and time integration scheme for each case.
tetrahedral mesh for beam with damping of rayleigh validation case
Figure 2: Tetrahedral mesh as used in case D.
hexahedral mesh for beam with damping of rayleigh validation case
Figure 3: Hexahedral mesh as used in case E.

Simulation Setup

Material:

  • Linear Elastic Isotropic with Damping of Rayleigh:
    • \(E = \) 35 \( GPa \)
    • \(\nu = \) 0.22
    • \(\rho = \) 2500 \( kg/m^3 \)
    • \(\alpha = \) 6.69e-5 \( 1/s \)
    • \(\beta = \) 20.06 \( s \)

Boundary Conditions:

  • Constraints:
    • Face ADEH is fixed
    • Face BCFG is fixed
  • Loads:
    • Linear pressure increment of 1e5 \(Pa\) on face EFGH until \(t = \) 1e-4 \( s \) and constant afterwards

Reference Solution

The reference solution is of numerical type as presented in cases A and B of [SDLS123]\(^1\).

Result Comparison

Comparison of the displacement (DZ), velocity (VZ), and acceleration (AZ) computed on the midpoint of the beam for cases A, B, and E.

displacement plot beam with damping of rayleigh validation case
Figure 4: Displacement results comparison
velocity plot beam with damping of rayleigh validation case
Figure 5: Velocity results comparison
acceleration plot beam with damping of rayleigh validation case
Figure 6: Acceleration results comparison

The results produced are in good agreement with the reference. Higher deviations are blamed on the nature of linear tetrahedral elements.

Following is a deformed shape of the beam at time \(t = \) 0.012 \(s\):

deformed shape and contour plot of beam with damping of rayleigh validation case
Figure 7: Deformed shape and contour plot at time 0.012 s.

Note

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

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