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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 solver in Simscale

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


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

    geometry validation beam harmonic
    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. The table below contains further information about the cases:

    CaseMesh TypeNodesElement TypeDampingLoad Type
    Case A1Standard839731st order tetrahedralNoTension
    Case A2Standard839731st order tetrahedralNoBending
    Case A3Standard839731st order tetrahedralRayleighTension
    Case A4Standard839731st order tetrahedralRayleighBending
    Case B1Standard5038672nd order tetrahedralNoTension
    Case B2Standard5038672nd order tetrahedralNoBending
    Case B3Standard5038672nd order tetrahedralRayleighTension
    Case B4Standard5038672nd order tetrahedralRayleighBending
    Table 2: Overview of the meshes and set up of the cases

    Find below the tetrahedral mesh used for case A:

    standard mesh harmonic analysis
    Figure 2: First-order mesh used for case A, created with the Standard meshing algorithm.

    Simulation Setup


    • 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 \(s\)
      • Beta damping = 0 \(1/s\)

    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\(^1\).


    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 3: 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
    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
    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
    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
    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 B4 in the post-processor:

    Real displacement on BEAM
    Figure 4: Harmonic analysis results for case B4, showing the displacement contours (magnified by a factor of 100).

    Last updated: June 14th, 2023