# Straight beam: Harmonic analysis¶

## Overview¶

The aim of this test case is to validate the following functions:

• Harmonic analysis solver

The simulation results of SimScale were compared to the results presented in [SHLL101]. Four mesh cases were considered; linear and quadratic tetrahedrals, linear and quadratic hexahedrals.

Import validation project into workspace

## Geometry¶

Geometry of a straight beam

## Analysis type and Domain¶

Tool Type : Code_Aster

Analysis Type : Harmonic analysis

Mesh and Element types :

Case Mesh type Number of nodes Number of 3D elements Element type
(A) linear tetrahedrals 159734 478774 3D isoparametric
(B) quadratic tetrahedrals 957963 478774 3D isoparametric
(C) linear hexahedrals 144240 72000 3D isoparametric
(D) quadratic hexahedrals 504600 72000 3D isoparametric

Mesh used for the SimScale Case A

Mesh used for the SimScale Case C

## Simulation Setup¶

Material:

• without damping:

• isotropic: E = 165.8 GPa, $$\nu$$ = 0.3, $$\rho$$ = 13404.106 kg/m³
• with damping:

• isotropic: E = 165.8 GPa, $$\nu$$ = 0.3, $$\rho$$ = 13404.106 kg/m³, $$\alpha$$ = 0.001 1/s, $$\beta$$ = 0 s,

Constraints:

• Red highlighted face constrained in all directions.

• tension:

• Force of 3000 N on green highlighted face in x direction.
• bending:

• Force of 3000 N on green highlighted face in y direction.

Simulation Control:

• Single frequency: 10 Hz

## Results¶

Comparison of the average displacement, velocity and acceleration on green highlighted face in x-direction (tension) or y-direction (bending) obtained with SimScale and the analytical results (within the assumption of Eulerian-Bernoulli beam theory) presented in [SHLL101].

• Tension:
Comparison of the Dx [m], Vx [m/s] and Ax [m/s²]
Quantity Material case [SHLL101] (real) [SHLL101] (imag) Case A (real) Case A (imag) Error [%] (real) Error [%] (imag) Case B (real) Case B (imag) Error [%] (real) Error [%] (imag) Case C (real) Case C (imag) Error [%] (real) Error [%] (imag) Case D (real) Case D (imag) Error [%] (real) Error [%] (imag)
Dx (m) without damping 5.318E-05 0 5.316E-05 0 0.03 0.00 5.317E-05 0 0.02 0.00 5.319E-05 0 0.01 0.00 5.317E-05 0 0.03 0.00
with damping 5.296E-05 -3.363E-06 5.295E-05 -3.363E-06 0.02 0.00 5.295E-05 -3.363E-06 0.01 0.00 5.297E-05 -3.364E-06 0.02 0.04 5.295E-05 -3.363E-06 0.01 0.00
Vx (m/s) without damping 0 3.341E-03 0 3.340E-03 0.00 0.02 0 3.341E-03 0.00 0.01 0 3.342E-03 0.00 0.02 0 3.341E-03 0.00 0.01
with damping 2.113E-04 3.327E-03 2.113E-04 3.327E-03 0.00 0.01 2.113E-04 3.327E-03 0.01 0.01 2.114E-04 3.328E-03 0.04 0.04 2.113E-04 3.327E-03 0.00 0.00
Ax (m/s²) without damping -2.099E-01 0 -2.099E-01 0 0.01 0.00 -2.099E-01 0 0.00 0.00 -2.100E-01 0 0.03 0.00 -2.099E-01 0 0.00 0.00
with damping -2.091E-01 1.327E-02 -2.090E-01 1.328E-02 0.03 0.04 -2.091E-01 1.328E-02 0.02 0.05 -2.091E-01 1.328E-02 0.01 0.09 -2.090E-01 1.328E-02 0.02 0.05
• Bending:
Comparison of the Dy [m], Vy [m/s] and Ay [m/s²]
Quantity Material case [SHLL101] (real) [SHLL101] (imag) Case A (real) Case A (imag) Error [%] (real) Error [%] (imag) Case B (real) Case B (imag) Error [%] (real) Error [%] (imag) Case C (real) Case C (imag) Error [%] (real) Error [%] (imag) Case D (real) Case D (imag) Error [%] (real) Error [%] (imag)
Dy (m) without damping -2.116E-02 0 -2.091E-02 0 1.21 0.00 -2.087E-02 0 1.38 0.00 -2.095E-02 0 1.00 0.00 -2.088E-02 0 1.37 0.00
with damping -2.102E-02 -1.823E-03 -2.077E-02 -1.783E-03 1.19 2.20 -2.073E-02 -1.774E-03 1.35 2.68 -2.081E-02 -1.790E-03 0.98 1.82 -2.074E-02 -1.775E-03 1.34 2.65
Vy (m/s) without damping 0 -1.330E+00 0 -1.314E+00 0.00 1.19 0 -1.311E+00 0.00 1.38 0 -1.316E+00 0.00 1.00 0 -1.312E+00 0.00 1.37
with damping 1.146E-01 -1.321E+00 1.120E-01 -1.305E+00 2.20 1.17 1.115E-01 -1.303E+00 2.68 1.35 1.125E-01 -1.308E+00 1.83 0.98 1.115E-01 -1.303E+00 2.65 1.34
Ay (m/s²) without damping 8.355E+01 0 8.256E+01 0 1.19 0.00 8.240E+01 0 1.38 0.00 8.272E+01 0 1.00 0.00 8.241E+01 0 1.37 0.00
with damping 8.298E+01 7.198E+00 8.201E+01 7.039E+00 1.17 2.20 8.285E+01 7.005E+00 0.15 2.68 8.216E+01 7.066E+00 0.98 1.83 8.186E+01 7.007E+00 1.34 2.65

## References¶

 [SHLL101] (1, 2, 3, 4, 5, 6) SHLL101 - Poutre droite. Analyse harmonique