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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: 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.

CaseMesh TypeNumber of NodesElement Type
A1st order Tetrahedral 191965Standard
B2nd order Tetrahedral189630Standard
C1st order Hexahedral10325Standard
D2nd order Hexahedral40935Standard
Table 1: Mesh details for each case
tetrahedral mesh shaft torque load validation case
Figure 2: Tetrahedral mesh used in case B
hexahedral mesh shaft torque load validation case
Figure 3: Hexahedral mesh used in case D

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 rotation angle

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

CASESIYZ\( \tau_{ref} \)ERROR
Table 3: Results comparison and computed error for shear stress
deformation color plot shaft torque load validation case
Figure 4: Deformation contour plot of the shaft

Tutorial: Linear Static Analysis of a Crane


  • (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: October 8th, 2020

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