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Validation Case: Bonded Contact on a Quarter Shaft

This bonded contact validation case belongs to solid mechanics. This test case aims to validate the following parameter:

  • Bonded contact

The simulation results obtained with SimScale were compared to the analytical results presented in [Roark]\(^1\).


Two geometries are used for this bonded contact validation. The first one consists of a quarter shaft. Radius is 0.1 \(m\) and length is 0.5 \(m\):

quarter shaft validation simscale
Figure 1: Quarter shaft consisting of a single part

The second geometry has the same dimensions, however, it is split exactly in half, between the ABC and A’B’C’ planes:

bonded contact validation shaft
Figure 2: Quarter shaft geometry, consisting of two parts

The 3D geometry is a 90\(^0\) section of a cylinder with dimensions as tabulated below:

Table 1: Geometry dimensions in meters

Analysis Type and Mesh

Tool Type: Code Aster

Analysis Type: Linear static

Mesh and Element Types: The meshes for Cases A through D were created in SimScale. The standard algorithm was used.

CaseGeometryMesh TypeNumber of NodesElement Type
(A)Quarter shaft1st order Standard8660Standard
(B)Quarter shaft – Split1st order Standard8846Standard
(C)Quarter shaft2nd order Standard62943Standard
(D)Quarter shaft – Split2nd order Standard63772Standard
Table 2: Mesh characteristics.

Find below the mesh used for Case D. It is a standard mesh with second-order tetrahedral cells.

tetrahedral second order mesh
Figure 3: Mesh used for Case D

Simulation Setup


  • Steel (linear elastic)
    • \(E\) = 208 \(GPa\), \(v\) = 0.3
    • Therefore, the shear modulus is 80 \(GPa\).

Boundary Conditions:

  • Constraints
    • Fixed support on face ABC
    • Face A’B’C’ is rotated with an angle \((\theta)\) of 2e-4 \(rad\)
  • Contacts
    • Single part shaft
      • Cyclic symmetry: face AA’B’B is tied to face AA’C’C. Rotation axis: z-axis. Sector angle: 90º
    • Split shaft
      • Cyclic symmetry: face AA”B”B is tied to face AA”C”C. Rotation axis: z-axis. Sector angle: 90º
      • Cyclic symmetry: face A”A’B’B” is tied to face A”A’C’C”. Rotation axis: z-axis. Sector angle: 90º
      • Bonded contact: both parts are bonded at A”B”C”.

Reference Solution

The analytical solution for maximum shear stress \(\tau_{max}\) given below is based on Roark\(^1\).

$$\large{\tau _{max}}=\frac {\theta.G.r}{l} = 3.2 [MPa]$$

Result Comparison

The results obtained from SimScale for the maximum shear stress \(\tau_{max}\) at point B’ are compared with the analytical solution by [Roark].

CaseQuantity[Roark]SimScaleError (%)
(A)Maximum shear stress \(\tau_{max} [MPa]\)3.23.209+0.281
(B)Maximum shear stress \(\tau_{max} [MPa]\)3.23.191-0.281
(C)Maximum shear stress \(\tau_{max} [MPa]\)3.23.195-0.156
(D)Maximum shear stress \(\tau_{max} [MPa]\)3.23.179-0.656
Table 3: Comparison of results at point B’.

Inspecting the Cauchy stress magnitude for Case B in the post-processor:

bonded contact validation cauchy stress
Figure 4: Case B (split geometry), showing Cauchy stress magnitude contours


  • W. C. YOUNG, R. G. BUDYNAS. Roark’s Formulas for Stress and Strain. Seventh Edition. McGraw-Hill. 2002. p. 415.
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