# Quarter shaft subjected to a fixed rotation¶

## Overview¶

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

• cyclic symmetry
• rotating motion

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 quarter shaft

The shaft has a radius $$r$$ = 0.1 m and a length $$l$$ = 0.5 m, but only one quarter is being modelled.

## Analysis type and Domain¶

Tool Type : Code_Aster

Analysis Type : Static

Mesh and Element types :

Case Mesh type Number of nodes Element type
(A) quadratic tetrahedral 2866 3D isoparametric
(B) quadratic tetrahedral 19243 3D isoparametric
(C) linear hexahedral 2730 3D isoparametric
(D) quadratic hexahedral 10527 3D isoparametric
 (a) Mesh used for the SimScale case (A) (b) Mesh used for the SimScale case (C)

## Simulation Setup¶

Material:

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

Constraints (Boundary Conditions):

• Face ABC is fixed
• Face A’B’C’ is rotated with an angle $$\theta$$ = 2*10⁻³ rad around the z-axis

Contacts:

• Cyclic symmetry, face AA’BB’ is tied to face AA’CC’
• rotation axis: z-axis
• Sector angle: 90°

## Reference Solution¶

$$$\label{ref1} \tau_{max} = \frac {\theta G r} {l} = 32 MPa$$$

## Results¶

Comparison of the maximum shear stress $$\tau_{max}$$ obtained with SimScale and the results derived from [Roark].

Comparison of maximum shear stress $$\tau_{max}$$ in [MPa] at point B
Case [Roark] SimScale Error
(A) 32 30.454 4.83%
(B) 32 32.1978 -0.62%
(C) 32 32 0.00%
(D) 32 32.059 -0.18%

## References¶

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