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Validation Case: Straight Beam with Damping of Rayleigh

This validation case belongs to the material damping of the Rayleigh model in solid mechanics. The aim of this test case is to validate the following parameters on the middle point of the beam:

  • Acceleration;
  • Velocity;
  • Displacement.

The simulation results from SimScale were compared to the Europlexus and Code Aster results presented in [SDLS123]\(^1\).


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 \([m]\)
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:

CaseMesh TypeNumber
of Nodes
Element Technology
AStandard11731st order tetrahedralStandard
BStandard74812nd order tetrahedralStandard
Table 2: Mesh details for each case
tetrahedral mesh for beam with damping of rayleigh validation case
Figure 2: Tetrahedral mesh as used in case A.

Simulation Setup


  • 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]\(^1\). The Europlexus and Code Aster results for the same linear triangular mesh were used for the comparison.

Result Comparison

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

displacement plot beam with damping of rayleigh validation case
Figure 3: Displacement results comparison
velocity plot beam with damping of rayleigh validation case
Figure 4: Velocity results comparison
acceleration plot beam with damping of rayleigh validation case
Figure 5: 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 for case A at time \(t = \) 0.012 \(s\):

deformed shape and contour plot of beam with damping of rayleigh validation case
Figure 6: Deformed shape and contour plot at time 0.012 \(s\) for case A, with a scaling factor of 1000


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

Last updated: September 28th, 2021

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