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

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

Mesh used for the SimScale case (A)

Mesh used for the SimScale case (C)

Simulation Setup

Material:

• isotropic: E = 208 GPa, ν = 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 τmax and the angle of twist θ obtained with SimScale and the results derived from [Roark].

Comparison of maximum shear stress τmax 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 θ 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

Last updated: January 29th, 2019

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