Validation Case: Circular Shaft Under Torque Load

This validation case belongs to solid mechanics. The aim of this test case is to validate the following parameters:

• Torque load applied through remote force boundary condition
• Shear stress distribution and maximum value
• Rotational deformation magnitude

The simulation results of SimScale were compared to the results presented in [Roark]\(^1\)

Geometry

The geometry used for the case is below:

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: Linear Static

Mesh and Element Types:

Tetrahedral meshes were computed using SimScale’s standard meshing algorithm and manual sizing. Table 1 shows an overview of the meshes in the validation project.

Case	Mesh Type	Number of Nodes	Element Type
A	Standard	191965	1st order tetrahedral
B	Standard	189630	2nd order tetrahedral

The second order tetrahedral mesh from case B is shown in Figure 2:

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):

\( \theta = ArcCos[ 1- \frac{U^2}{2R^2} ] \tag{4} \)

CASE	\(U\)	\( \theta \)	\( \theta_{ref} \)	ERROR
A	0.000198463	0.00198463	0.0019894	-0.24%
B	0.000198944	0.00198944	0.0019894	0.00%

The maximum shear stress is taken from the Cauchy stress tensor, component [SIYZ]:

CASE	SIYZ	\( \tau_{ref} \)	ERROR
A	31.7483	31.847	– 0.26%
B	31.8405	31.847	– 0.03%

For both 1st order and 2nd order meshes, the results show good agreement with the analytical solution. Find below an image showing the displacement magnitude on the cylinder, for case B:

Related Tutorials

• Linear Static Analysis of a Crane

Last updated: July 21st, 2021