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]\(^1\).

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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:

POINT	X \([m]\)	Y \([m]\)	Z \([m]\)
A	0	0	0
B	10	0	0
C	10	1	0
D	0	1	0
E	0	0	1
F	10	0	1
G	10	1	1
H	0	1	1

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:

Case	Mesh Type	Number of Nodes	Element Type	Element Technology
A	Standard	1173	1st order tetrahedral	Standard
B	Standard	7481	2nd order tetrahedral	Standard

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

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