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Circular Shaft Under Torque

Overview

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

  • torque
  • remote force

The simulation results of SimScale were compared to the numerical results presented in [Roark]. The meshes used in (A) and (B) were created with the parametrized-tetrahedralization-tool on the SimScale platform. The meshes used in (C) and (D) were locally created with Salome.

Import validation project into workspace

Geometry

CircularShaftUnderTorque-geometry

Geometry of the shaft

The shaft has a radius \(r\) = 0.1 m and a length of \(l\) = 0.5 m.

Analysis type and Domain

Tool Type : CalculiX/Code_Aster

Analysis Type : Static

Mesh and Element types :

Case Mesh type Number of nodes Element type
(A) linear tetrahedral 12940 3D isoparametric
(B) quadratic tetrahedral 94919 3D isoparametric
(C) linear hexahedral 10325 3D isoparametric
(D) quadratic hexahedral 40935 3D isoparametric
CircularShaftUnderTorque-mesh-a

Mesh used for the SimScale case (A)

CircularShaftUnderTorque-mesh-c

Mesh used for the SimScale case (C)

Simulation Setup

Material:

  • isotropic: E = 208 GPa, \(\nu\) = 0.3, G = 80 GPa

Constraints:

  • Face A is fixed

Loads:

  • Torque \(T\) of 50000 N/m on face B

Reference Solution

\[\begin{equation}\label{ref1} J = \frac{1}{2} \pi r^4 = 1.57 \cdot 10^{-4} m^4 \end{equation}\]\[\begin{equation}\label{ref2} \tau_{max} = \frac {T r} {J} = 31.847 N/mm^2 \end{equation}\]\[\begin{equation}\label{ref3} \theta = \frac {\tau_{max} l} {G r} = 1.9904 \cdot 10^{-4} rad \end{equation}\]

Results

Important

  • The analytical solution assumes an undeformable surface on the face, wich is subject to the torque bc. This can be modelled in Code_Aster with the option ‘undeformable’ in the remote displacement bc.
  • CalculiX has no such option and therefore the stresses in the entitites, which are assigned to the torque bc, are unphysical in CalculiX. So the stresses were computed at a point on the edge of face A.

Comparison of the maximum shear stress \(\tau_{max}\) and the angle of twist \(\theta\) obtained with SimScale and the results derived from [Roark].

Comparison of maximum shear stress \(\tau_{max}\) in [MPa]
Case Tool Type [Roark] SimScale Error
(A) CalculiX 31.847 30.212 5.13%
(A) Code_Aster 31.847 30.212 5.13%
(B) CalculiX 31.847 31.842 0.02%
(B) Code_Aster 31.847 31.838 0.03%
(C) Code_Aster 31.847 32.066 -0.69%
(D) Code_Aster 31.847 31.879 -0.10%
Comparison of the angle of twist \(\theta\) in [rad]
Case Tool Type [Roark] SimScale Error
(A) CalculiX 0.0019904 0.001953 1.88%
(A) Code_Aster 0.0019904 0.001969 1.08%
(B) CalculiX 0.0019904 0.001969 1.08%
(B) Code_Aster 0.0019904 0.001989 0.07%
(C) Code_Aster 0.0019904 0.002004 -0.68%
(D) Code_Aster 0.0019904 0.00199 0.02%

References

[Roark](1, 2, 3, 4) (2011)”Roark’s Formulas For Stress And Strain, Eighth Edition”, W. C. Young, R. G. Budynas, A. M. Sadegh