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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]\(^1\)


The geometry used for the case is below:

geometrical model shaft torque load validation case
Figure 1: Geometrical model of the shaft

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.

CaseMesh TypeNumber of NodesElement Type
AStandard 1919651st order tetrahedral
BStandard1896302nd order tetrahedral
Table 1: Mesh details for each case

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

tetrahedral mesh shaft torque load validation case
Figure 2: Shaft geometry meshes with 2nd order tetrahedral cells

Simulation Setup


  • 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
Table 2: Results comparison and computed error for the rotation angle

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

CASESIYZ\( \tau_{ref} \)ERROR
A31.748331.847– 0.26%
B31.840531.847– 0.03%
Table 3: Results comparison and computed error for shear stress

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:

deformation color plot shaft torque load validation case
Figure 3: Case B results, showing the displacement magnitude contours


  • (2011) “Roark’s Formulas For Stress And Strain, Eighth Edition”, W. C. Young, R. G. Budynas, A. M. Sadegh.


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Last updated: July 21st, 2021