Torsion of a Beam with the Square Cross-Section

Let us consider a beam with the square cross-section. The length of the side of a square is a. Length of the beam is L (see figure).

The beam is subjected to the externally applied torque Mt. The torque is applied at the right end of the beam, the left end of the beam is rigidly clamped.

Torsion of a Beam with the Square Cross-Section, the finite element model with applied loads and restraints

The finite element model with applied loads and restraints

Let us use the following initial data: length L of the beam is 1.5 m, the length of the side of a square a is 0.050 m, the magnitude of the applied torque Mt is 1000 N-m.
Material characteristics: E = 2.0E+011 Pa, ν = 0.29.
To find the angle of twist, let us use the following relation:
,
where G=E/2(1+ν) – shear modulus, Jp=βa4 – polar moment of inertia of the square cross-section, β= 0.1406.
Thus, ϕ= 2.2168E-002 rad.
The maximal displacement is calculated by the following formula:

Thus, Δu = 7.8371E-004 m.
The maximal shear stress τ max is calculated by the following formula:

where α= 0.208
Thus, τ max = 3.8462E+007 Pa.
After carrying out calculation with the help of AutoFEM, the following results are obtained:

Table 1.Parameters of finite element mesh

Finite Element Type	Number of nodes	Number of Finite Elements
quadratic tetrahedron	24449	14862

Table 2. Result "Displacement"

Numerical Solution Displacement Δu*, m	Analytical Solution Displacement Δu, m	Error δ = 100%* \| Δu* - Δu \| / \| Δu \|
7.7619E-004	7.8371E-004	0.96

Table 3. Result "Shear Stress"

Numerical Solution Shear stress τyz max*, Pa	Analytical Solution Shear stress τmax, Pa	Error δ = 100%\| τmax - τmax \| / \| τmax \|
4.0144E+007	3.8462E+007	4.37

Torsion of a Beam with the Square Cross-Section, Result "Displacement"

Conclusions:

The relative error of the numerical solution compared to the analytical solution is 0.96% for displacements and 4.37% for stresses when using quadratic finite elements.

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