The axis of the shaft cylinder is aligned with the \(Z\) axis, with a length \(L = \) 0.5 \(m\) and a radius \(r = \) 0.1 \(m\).
Analysis Type and Mesh
Tool Type: Code_Aster
Analysis Type: Static Linear
Mesh and Element Types:
Tetrahedral meshes were computed using SimScale’s standard mesh algorithm and manual sizing. Hexahedral meshes were locally computed and imported into the project.
Case
Mesh Type
Number of Nodes
Element Type
A
1st order Tetrahedral
191965
Standard
B
2nd order Tetrahedral
189630
Standard
C
1st order Hexahedral
10325
Standard
D
2nd order Hexahedral
40935
Standard
Table 1: Mesh details for each case
Figure 2: Tetrahedral mesh used in case B
Figure 3: Hexahedral mesh used in case D
Simulation Setup
Material:
Linear Elastic Isotropic:
\(E = \) 208 \(GPa\)
\(\nu = \) 0.3
\(G = \) 80 \(GPa \)
Boundary Conditions:
Constraints:
Face A is fixed.
Loads:
Torque \(T = \) 50000 \(Nm\) on face B.
Reference Solution
The analytical solutions for the rotation angle \(\theta_B\) and maximum shear stress \(\tau_{max}\) are given by the following equations:
\( \theta_B = \frac{ T L }{G J} \tag{1} \)
\( \tau_{max} = \frac{T R}{J} \tag{2} \)
\( J = \frac{\pi R^4}{2} \tag{3} \)
The computed reference solution is:
\( \theta_B = 1.9894×10^{-3}\ Rad \)
\( \tau_{max} = 31.847\ MPa \)
Result Comparison
The plane maximum displacement \( U \) is used to compute the rotation angle through equation 4 (obtained through the cosines law):
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