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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]$$^1$$

## Geometry

The geometry used for the case is below:

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: Linear Static

Mesh and Element Types:

Tetrahedral meshes were computed using SimScale’s standard meshing algorithm and manual sizing. Table 1 shows an overview of the meshes in the validation project.

The second order tetrahedral mesh from case B is shown in Figure 2:

## Simulation Setup

Material:

• Linear Elastic Isotropic:
• $$E =$$ 208 $$GPa$$
• $$\nu =$$ 0.3
• $$G =$$ 80 $$GPa$$

Boundary Conditions:

• Constraints:
• Face A is fixed.
• 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):

$$\theta = ArcCos[ 1- \frac{U^2}{2R^2} ] \tag{4}$$

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

For both 1st order and 2nd order meshes, the results show good agreement with the analytical solution. Find below an image showing the displacement magnitude on the cylinder, for case B: