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    Circular Shaft Under Torque


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

    • torque
    • remote force

    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.

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    Geometry of the shaft

    The shaft has a radius r = 0.1 m and a length of l = 0.5 m.

    Analysis type and Domain

    Tool Type : CalculiX/Code_Aster

    Analysis Type : Static

    Mesh and Element types :

    Case Mesh type Number of nodes Element type
    (A) linear tetrahedral 12940 3D isoparametric
    (B) quadratic tetrahedral 94919 3D isoparametric
    (C) linear hexahedral 10325 3D isoparametric
    (D) quadratic hexahedral 40935 3D isoparametric

    Mesh used for the SimScale case (A)


    Mesh used for the SimScale case (C)

    Simulation Setup


    • isotropic: E = 208 GPa, ν = 0.3, G = 80 GPa


    • Face A is fixed


    • Torque T of 50000 N/m on face B

    Reference Solution

    J = \frac{1}{2} \pi r^4 = 1.57 \cdot 10^{-4} m^4
    \tau_{max} = \frac {T r} {J} = 31.847 N/mm^2
    \theta = \frac {\tau_{max} l} {G r} = 1.9904 \cdot 10^{-4} rad



    • The analytical solution assumes an undeformable surface on the face, wich is subject to the torque bc. This can be modelled in Code_Aster with the option ‘undeformable’ in the remote displacement bc.
    • CalculiX has no such option and therefore the stresses in the entitites, which are assigned to the torque bc, are unphysical in CalculiX. So the stresses were computed at a point on the edge of face A.

    Comparison of the maximum shear stress τmax and the angle of twist θ obtained with SimScale and the results derived from [Roark].

    Comparison of maximum shear stress τmax in [MPa]
    Case Tool Type [Roark] SimScale Error
    (A) CalculiX 31.847 30.212 5.13%
    (A) Code_Aster 31.847 30.212 5.13%
    (B) CalculiX 31.847 31.842 0.02%
    (B) Code_Aster 31.847 31.838 0.03%
    (C) Code_Aster 31.847 32.066 -0.69%
    (D) Code_Aster 31.847 31.879 -0.10%
    Comparison of the angle of twist θ in [rad]
    Case Tool Type [Roark] SimScale Error
    (A) CalculiX 0.0019904 0.001953 1.88%
    (A) Code_Aster 0.0019904 0.001969 1.08%
    (B) CalculiX 0.0019904 0.001969 1.08%
    (B) Code_Aster 0.0019904 0.001989 0.07%
    (C) Code_Aster 0.0019904 0.002004 -0.68%
    (D) Code_Aster 0.0019904 0.00199 0.02%


    [Roark] (1234) (2011)”Roark’s Formulas For Stress And Strain, Eighth Edition”, W. C. Young, R. G. Budynas, A. M. Sadegh

    Last updated: January 29th, 2019

    What's Next

    part of: Compressible Flow: de Laval Nozzle