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:
The simulation results obtained with SimScale were compared to the analytical results presented in [Roark]\(^1\).
Geometry 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 \(m\):
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:
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:
A B C A’ B’ C’ A” B” C” x 0 0.1 0 0 0.1 0 0 0.1 0 y 0 0 0.1 0 0 0.1 0 0 0.1 z 0 0 0 0.5 0.5 0.5 0.25 0.25 0.25
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.
Case Geometry Mesh Type Number of Nodes Element Type (A) Quarter shaft 1st order Standard 8660 Standard (B) Quarter shaft – Split 1st order Standard 8846 Standard (C) Quarter shaft 2nd order Standard 62943 Standard (D) Quarter shaft – Split 2nd order Standard 63772 Standard
Table 2: Mesh characteristics. Find below the mesh used for Case D. It is a standard mesh with second-order tetrahedral cells.
Figure 3: Mesh used for Case D Simulation Setup Material :
Steel (linear elastic)\(E\) = 208 \(GPa\), \(v\) = 0.3 Therefore, the shear modulus is 80 \(GPa\). Boundary Conditions :
ConstraintsFixed support on face ABC Face A’B’C’ is rotated with an angle \((\theta)\) of 2e-4 \(rad\) ContactsSingle part shaftCyclic symmetry: face AA’B’B is tied to face AA’C’C. Rotation axis: z-axis. Sector angle: 90º Split shaftCyclic 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].
Case Quantity [Roark] SimScale Error (%) (A) Maximum shear stress \(\tau_{max} [MPa]\) 3.2 3.209 +0.281 (B) Maximum shear stress \(\tau_{max} [MPa]\) 3.2 3.191 -0.281 (C) Maximum shear stress \(\tau_{max} [MPa]\) 3.2 3.195 -0.156 (D) Maximum shear stress \(\tau_{max} [MPa]\) 3.2 3.179 -0.656
Table 3: Comparison of results at point B’. Inspecting the Cauchy stress magnitude for Case B in the post-processor:
Figure 4: Case B (split geometry), showing Cauchy stress magnitude contours References
W. C. YOUNG, R. G. BUDYNAS. Roark’s Formulas for Stress and Strain. Seventh Edition. McGraw-Hill. 2002. p. 415. Last updated: June 24th, 2020
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