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]\(^1\).
The geometry used for the case is as follows:
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. Hexahedral meshes were locally computed and imported into the project.
Case Mesh Type Number of Nodes Element Type Integration Scheme A 1st order Hexahedral 496 Standard Explicit B 2nd order Hexahedral 1720 Standard Explicit C 1st order Tetrahedral 2862 Standard Explicit D 2nd order Tetrahedral 1575 Standard Explicit E 2nd order Hexahedral 1720 Standard Implicit F 2nd order Tetrahedral 1575 Standard Implicit Table 2: Mesh details and time integration scheme for each case.
Figure 2: Tetrahedral mesh as used in case D.
Figure 3: Hexahedral mesh as used in case E. 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 \( s \)
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]\(^1\).
Comparison of the displacement (DZ), velocity (VZ), and acceleration (AZ) computed on the midpoint of the beam for cases A, B, and E.
Figure 4: Displacement results comparison
Figure 5: Velocity results comparison
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 \(s\):
Figure 7: Deformed shape and contour plot at time 0.012 s.
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