Cross-Section Classification in Case of Uniaxial Bending with Axial Force

Technical Article

The RF-/STEEL EC3 add‑on module performs a detailed cross‑section classification at each design before the design is carried out. Thus, the susceptibility to local buckling of all cross‑section parts is evaluated. The defined cross-section class has an effect on the resistance and rotational capacity determination.

Cross-Section Classes

Eurocode 3 [1] specifies four classes of cross‑sections:

Figure 01 - Cross-Section Classes

The cross‑section classification provides the following parameters and boundary conditions:

  • Support of a cross‑section part (on one or both sides)
  • Length c of a cross‑section part
  • Thickness t of a cross‑section part
  • Yield strength of the steel used in the form of the factor ε
  • Distribution of stresses on the designed cross‑section part

The class of the cross‑section part with the least favourable value is governing for the entire cross‑section. For I‑sections and H‑sections, this is usually the comparatively slender web.

The stress distribution is detected by the parameter α (plastic, Class 1 and 2) or ψ (elastic, Class 3). In this case, α represents the percentage length of the compression stress in the cross‑section part while ψ represents the ratio of the boundary stresses.

Figure 02 - Explanation of α and ψ

Important Note:

  • The existing stresses are always calculated up or down to the yield strength.
  • Compression stresses are always to be set as positive, tensile stresses as negative.

In solely uniaxial bending in a double symmetric cross‑section, the determination of α and ψ is trivial. An additional axial force requires further considerations. The interesting question is, to what extent does the normal force apply? There are two approaches and both are implemented in the RF‑/STEEL EC3 add‑on module.

Figure 03 - Types of Determination of α and ψ

First, there is the second option ‘Increase NEd and MEd uniformly’, which is preset in RF‑/STEEL EC3. In the case of the elastic stress distribution, the existing stresses are increased by the yield stress/largest compression stress ratio in the cross‑section part. The ψ parameter results from the relation of compression stress and tension stress. If the stress distribution is plastic, the moment and axial force is increased until one of the interaction conditions specified in [1] is reached and thus the plastic limit state is reached. See the explanation in [2], page 13.

RF-/STEEL EC3 uses the interaction condition according to Formula 6.2 because it is easily traceable and valid for all types of cross‑sections. The following graphic shows an example of IPE 360, S 235, with the following internal forces and plastic resistances:

My,Ed = 125.0 kNm    NEd =    300.0 kN
My,Rd = 239.5 kNm    NRd = 1,709.0 kN

Figure 04 - Interaction Diagram

Extrapolation of the existing stresses results in the following limit internal forces:

MN,y,Rd = 179.2 kNm    NMy,Rd = 430.1 kN

On the basis of the limit axial force, the size of the stress block can now be determined and applied to in the area bisecting axes of the cross‑section. By considering the remaining stress blocks of the bending moment, you can now determine the length of the compression stress in the cross‑section part and thus the parameter α.

Figure 05 - Calculation of α

The first option ‘Fixed NEd, increase MEd to reach fyd’ can be easily explained on the plastic stress distribution.The axial force is not extrapolated but rather applied to the acting size. As a result, the compression area and α are generally smaller when using this option.

The determination of c/t limit values for the individual cross‑section classes are not further explained in this article. This information can be found in [1], Table 5.2.

References

[1]   Eurocode 3: Design of steel structures - Part 1‑1: General rules and rules for buildings; EN 1993‑1‑1:2005 + AC:2009
[1]   SEMI-COMP+: Berechnungsrichtlinie für die Querschnitts- und Stabbemessung nach Eurocode 3 mit Schwerpunkt auf semi-kompakten Querschnitten. (2011). Graz: TU Graz - Institut für Stahlbau.

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