C10. Specific behavior in shear and torsion

C10. Specific behavior in shear and torsion

In general, it should be noted that beam element models do not systematically consider shear stress deformations, nor do they adequately account for torsional deformations.

However, in the case of modeling a structure that is sensitive to shear and torsion, one must activate the option to consider shear and torsional deformations and to clearly define the reduced cross-sections and torsional inertias.

It can also be noted that the phenomena with blocked torsion are impossible to model in beam-element structures because the beam elements of Strength of Materials are built on the assumption of conservation of straight sections (without distortion or buckling) and yet, their consideration leads to stress distributions different from those calculated in "classical" Strength of Materials.

Considering the blocked torsion will generally require the separate modeling of all the plates constituting the thin profile of the section.

Here are some examples of structures sensitive to these phenomena:

• for shear: slender welded beams (mixed double girders, for example), console-type structure (bracing walls with a low height/length ratio). Failure to consider the shear deformation will result either in an underestimation of the deformations leading to an erroneous deflection or an overestimation of the stiffnesses,

• for torsion: structures not free to distort (at one or several points).

Comparing calculations of the angle of rotation of a cantilever I-beam

Data - cantilever beam:

• Boundary conditions: fixed in x=0 (θ=0, θ'=0) and free in x=L (B=0, T=0)

• Length: L = 1 m

• Loading: torsional moment at the end x=L: M_x = 10 kN.m

• Cross-section: Welded h_t = 200 mm, b_t = 200 mm, t_f = 20 mm, t_w = 10 mm

Plate element model: 

Loading: 

Reaction: 

Displacement θ(L)=0.042 rad

Beam-element model 

Loading:

Reaction: 

Displacement θ(L)=0.1198 rad

Analytical calculation

The differential equation for the angle of rotation is given by:

With the boundary conditions given in the previous paragraph, the solution of this equation is:

With:

• I_t: St Venant torsional inertia 

• I_ω: sectoral inertia

• M_x: torsional moment

• L: length of the beam

• 

Application:

• L = 1 m

• G = 80,770 MPa

• E = 210,000 MPa

• (calculated by the software)

• (calculated by the software)

• M_x = 10 kN.m

• 

The analytical calculation and the surface element model give the same rotation result θ(L)=0.042 rad.

The beam element model calculation gives a result 2.85 times higher.

In the beam element model, the stiffness due to the buckling inertia is not taken into account for the calculation of the rotation angle:

Conclusion

In general, for beam element models, the stiffnesses due to the torsion of an open section beam are not considered properly in the calculations.

In case of any doubt, a shell-type element approach on a simplified, global, or local model can help identifying the effects.