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]$^1$.

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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 $m$:

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:

	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.

tetrahedral second order mesh — 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:

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].

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:

bonded contact validation cauchy stress — 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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Documentation

Validation Case: Bonded Contact on a Quarter Shaft

Geometry

Analysis Type and Mesh

Simulation Setup

Reference Solution

Result Comparison

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Validation Case: Bonded Contact on a Quarter Shaft

Geometry

Analysis Type and Mesh

Simulation Setup

Reference Solution

Result Comparison

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