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Validation Case: Harmonic Analysis of a Straight Beam

This harmonic analysis of a straight beam validation case belongs to solid mechanics. It aims to validate the following parameter:

  • Harmonic analysis

The harmonic analysis results from SimScale were compared to the numerical results in SHLL101\(^1\).

Geometry

Find below the straight beam geometry used for this validation case:

Figure 1: Geometry used in the present validation case

The dimensions of the geometry are tabulated below. Furthermore, the center point C is located at the coordinates (0, 0, 0):

Geometry ParameterDimension \([m]\)
Outer radius \((R)\)0.0925
Inner radius \((r)\)0.08638
Length \((L)\)10
Table 1: Dimensions of the straight beam

Analysis Type and Mesh

Tool Type: Code_Aster

Analysis Type: Harmonic analysis

Mesh and Element Types: A total of four meshes are used in this validation case. The standard algorithm was used to create the meshes for cases A and B. Two hexahedral meshes were created locally and imported to SimScale for cases C and D. The table below contains further information about the cases:

CaseMesh TypeNodesElement TypeDampingLoad Type
Case A11st order standard83973StandardNoTension
Case A21st order standard83973StandardNoBending
Case A31st order standard83973StandardRayleighTension
Case A41st order standard83973StandardRayleighBending
Case B12nd order standard503867StandardNoTension
Case B22nd order standard503867StandardNoBending
Case B32nd order standard503867StandardRayleighTension
Case B42nd order standard503867StandardRayleighBending
Case C11st order hexahedral144240StandardNoTension
Case C21st order hexahedral144240StandardNoBending
Case C31st order hexahedral144240StandardRayleighTension
Case C41st order hexahedral144240StandardRayleighBending
Case D12nd order hexahedral504600StandardNoTension
Case D22nd order hexahedral504600StandardNoBending
Case D32nd order hexahedral504600StandardRayleighTension
Case D42nd order hexahedral504600StandardRayleighBending
Table 2: Overview of the meshes and set up of the cases

Find below the hexahedral mesh used for case C:

hexahedral mesh harmonic analysis
Figure 2: First-order hexahedral mesh used for case C

Figure 3 shows the second-order standard mesh used for case D:

standard mesh harmonic analysis validation
Figure 3: Second-order standard mesh used in case D

Simulation Setup

Material:

  • Material behavior: Linear elastic
  • \((E)\) Young’s modulus = 165.8 \(GPa\)
  • \((v)\) Poisson’s ratio = 0.3
  • \((\rho)\) Density = 13404.106 \(kg/m^3\)
  • Rayleigh damping settings (for the cases that model damping):
    • Alpha coefficient = 0.001 \(1/s\)
    • Beta damping = 0 \(s\)

Boundary Conditions:

  • Constraint:
    • The free end of the cylinder at \(x\) = 0 is a Fixed support
  • Load:
    • The other free end of the cylinder, at \(x\) = 10 \(m\), receives a bending or a tension load, according to table 2.
      • Tension load: Force boundary condition, with \(F_x\) = 3000 \(N\)
      • Bending load: Force boundary condition, with \(F_y\) = 3000 \(N\)

Reference Solution

The analytical solution for the displacements, velocity, and acceleration on the point (10, 0, 0) are presented in SHLL101\(^1\).

Note

The point (10, 0, 0), where the reference results were calculated, is the center point of the annulus.

As this point is not within the solid walls of the pipe, a total of 4 points were created on the free end to assess the results:

data point placement
Figure 4: Points at the free end (\(x\) = 10) where data was collected

The coordinates of the points are as follows:
– Left: (10, 0, 0.08944)
– Right: (10, 0, -0.08944)
– Top: (10, 0.08944, 0)
– Bottom: (10, -0.08944, 0)

SimScale results are the average of the values obtained at these four points. When tension is being applied, the results are evaluated in the x-direction. Otherwise, if a bending load is applied, the results are taken in the y-direction.

Result Comparison

In Table 3, we compare SimScale results with the reference SHLL101\(^1\), modeling tension load, and no damping.

CaseDisplacement in x-direction [\(m\)]Error [%]Velocity in x-direction [\(m/s\)]Error [%]Acceleration in x-direction [\(m/s^2\)]Error [%]
SHLL101 (Real)5.3180E-050-2.0990E-01
SHLL101 (Imaginary)03.3410E-030
Case A1 (Real)5.3433E-050.473500-2.1095E-010.4973
Case A1 (Imaginary)003.3580E-030.506000
Case B1 (Real)5.3105E-05-0.140900-2.0965E-01-0.1188
Case B1 (Imaginary)003.3367E-03-0.129200
Case C1 (Real)5.3185E-050.009800-2.0997E-010.0319
Case C1 (Imaginary)003.3417E-030.021500
Case D1 (Real)5.3173E-05-0.014100-2.0992E-010.0081
Case D1 (Imaginary)003.3509E-030.296300
Table 3: Result comparison – tension load and no damping

Table 4 shows a similar comparison, but now for the cases with a bending load and no damping.

CaseDisplacement in y-direction [\(m\)]Error [%]Velocity in y-direction [\(m/s\)]Error [%]Acceleration in y-direction [\(m/s^2\)]Error [%]
SHLL101 (Real)-2.1160E-0208.3550E+01
SHLL101 (Imaginary)0-1.3300E+000
Case A2 (Real)-2.1066E-02-0.4443008.3167E+01-0.4606
Case A2 (Imaginary)00-1.3236E+00-0.480500
Case B2 (Real)-2.1066E-02-0.4443008.3167E+01-0.4606
Case B2 (Imaginary)00-1.3237E+00-0.479700
Case C2 (Real)-2.0951E-02-0.9990008.2710E+01-1.0156
Case C2 (Imaginary)00-1.3164E+00-1.035400
Case D2 (Real)-2.0875E-02-1.3643008.2412E+01-1.3809
Case D2 (Imaginary)00-1.3116E+00-1.400500
Table 4: Result comparison – bending load and no damping

Now, comparing the cases with tension load and the Rayleigh damping model:

CaseDisplacement in x-direction [\(m\)]Error [%]Velocity in x-direction [\(m/s\)]Error [%]Acceleration in x-direction [\(m/s^2\)]Error [%]
SHLL101 (Real)5.2960E-052.1130E-04-2.0910E-01
SHLL101 (Imaginary)-3.3630E-063.3270E-031.3270E-02
Case A3 (Real)5.3219E-050.48592.1237E-040.5038-2.1010E-010.4764
Case A3 (Imaginary)-3.3800E-060.50213.3439E-030.50451.3344E-020.5516
Case B3 (Real)5.3226E-050.49982.1262E-040.6208-2.1007E-010.4603
Case B3 (Imaginary)-3.3758E-060.38033.3433E-030.48811.3359E-020.6684
Case C3 (Real)5.2972E-050.02172.1138E-040.0378-2.0912E-010.0110
Case C3 (Imaginary)-3.3642E-060.03633.3283E-030.03911.3281E-020.0858
Case D3 (Real)5.2959E-05-0.00212.1133E-040.0142-2.0907E-01-0.0129
Case D3 (Imaginary)-3.3634E-060.01283.3275E-030.01531.3278E-020.0625
Table 5: Result comparison – tension load with damping

Lastly, the results for bending load and damping are compared in Table 6:

CaseDisplacement in y-direction [\(m\)]Error [%]Velocity in y-direction [\(m/s\)]Error [%]Acceleration in y-direction [\(m/s^2\)]Error [%]
SHLL101 (Real)-2.1020E-021.1460E-018.2980E+01
SHLL101 (Imaginary)-1.8230E-03-1.3210E+007.1980E+00
Case A4 (Real)-2.0924E-02-0.45831.1325E-01-1.19038.2605E+01-0.4537
Case A4 (Imaginary)-1.8025E-03-1.1390-1.3147E+00-0.47847.1159E+00-1.1542
Case B4 (Real)-2.0924E-02-0.45831.1325E-01-1.18948.2605E+01-0.4537
Case B4 (Imaginary)-1.8025E-03-1.1390-1.3147E+00-0.47847.1159E+00-1.1542
Case C4 (Real)-2.0808E-02-1.01791.1247E-01-1.89208.2154E+01-1.0057
Case C4 (Imaginary)-1.7901E-03-1.8407-1.3075E+00-1.03107.0668E+00-1.8563
Case D4 (Real)-2.0737E-02-1.36571.1154E-01-2.74068.1865E+01-1.3616
Case D4 (Imaginary)-1.7753E-03-2.6886-1.3029E+00-1.38697.0085E+00-2.7042
Table 6: Result comparison – bending load with damping

SimScale results show great agreement with the analytical solution for all configurations.

Inspecting the displacement magnitude for case D4 in the post-processor:

displacement contours post processor
Figure 5: Harmonic analysis results for case D4, showing the displacement contours (magnified by a factor of 100)

Last updated: August 13th, 2020

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