Torsion of a shaft under the action of two torques

Let us consider a shaft of length L, diameter d, loaded with two torques T of inverse directions, applied to either of the ends. The cross-section of the shaft is a circle of diameter d.

Sought quantity is the maximum angle of twisting.
Assume T = 100 N*m, L = 0.5 m, d = 0.06 m.
Characteristics of material have values: Shear modulus G = 8.2E+010 Pa, Poisson's ratio ν = 0.28. Points at centers of both ends are partially restricted in normal-to-axis directions (those restrictions are still not sufficient to determine model in space and therefore to complete calculation - this is done to facilitate examining results). Both ends are loaded with torque T, pointing in opposite directions to each other.
To solve this study in AutoFEM Analysis, it is necessary to turn on the option "Stabilize the unfixed model" with additional stiffness equal 1.
You should check this box at the Properties dialog of Static Analysis on the page "Solving".

The analytical solution appears as:
 φ = (T*L)/(G*Jp) = 4.791E-004 rad
w = d*sin(φ/2) = 1.437E-005 m

where φ - angle of twist, w - point displacement, T – is the torque, L – the shaft length, G – the material Shear modulus,

        Jp= πd4 / 32 – polar moment of inertia of the circular cross-section.

After carrying out calculation with the help of AutoFEM, the following results are obtained:

(shaft deflection is equal to (7.388E-006)+(7.375E-006)=1.476E-006 m)

Table 1.Parameters of the finite element mesh

Table 2. Result "Displacement, magnitude"*

Conclusions:

The relative error of the numerical solution compared to the analytical solution is equal to 2.7% for quadratic finite elements.

*The results of numerical tests depend on the finite element mesh and may differ slightly from those given in the table.

 

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