Cross-Section Classification and Plastic Analysis in SHAPE-THIN
The cross‑section properties software SHAPE‑THIN determines effective section properties of thin‑walled cross‑sections according to Eurocode 3 and Eurocode 9. Alternatively, the program allows for plastic design of general cross‑sections according to Simplex Method. In this process, plastic cross‑section reserves are iteratively calculated for elastically determined internal forces.
The following example describes the effective cross‑section properties in the notching area of a rolled I‑section. Afterwards, the results are compared with the plastic analysis.
The cross‑section is created using the cross‑section library. When setting the rolled section IPE 200, the ‘Reduce section into individual elements’ option is activated. The material of the section is structural steel S 235.
The notch can be generated by dividing the web element at a distance of 30 mm. Then, you can remove the overlapping elements and fillets.
In our example, a quite large compression force with small bending moments is specified.
Constellations of internal forces can be analysed using diverse load cases, x‑locations, or member numbers. In addition, SHAPE‑THIN provides the option to import the internal forces from RFEM or RSTAB.
After the calculation, Cross‑Section Class 1 is assigned to the flanges. The web belongs to Cross‑Section Class 3.
In order for the web to reach the yield strength for an elastic stress distribution, the plastic moment resistance cannot develop due to local buckling. However, the local buckling does not occur before reaching the yield strength.
As you can see in the stresses graphic, the bending moment My reduces the compressive stresses at the free edge of the web. For further design, this moment is set to zero. The modified constellation of internal forces can be assigned to a new member, for example.
As a result, the web is now assigned to Cross‑Section Class 4. The effective width of the cross‑sectional part is only 62% of the member length.
The stresses can be displayed both on the full cross‑section (on top left in the figure) and on the reduced cross‑section (right).
In SHAPE-THIN, it is also possible to perform the plastic analysis for both constellations of internal forces. For this, it is necessary to select the ‘Plastic capacity design’ option in the General Data dialog box. In this design, the enlargement factor αplast is determined as the maximum of a linear optimisation task in order to reach the plastic cross‑section resistance, taking into account the interaction conditions (‘Revised Simplex Algorithm’).
The simplex calculation gives the result of plastic reserves of 541% and 862% for both member internal forces.
It is apparent that the cross‑section resistance is far higher in the case of compression (Member 2) than in the case of the combined effects, although only a part of the web is effective according to Eurocode 3. However, in this analysis, the different approaches of the two methods must not be mixed; according to Eurocode 3, it is disputable whether a cross‑section part is prone to buckling. This is the case of the Class 4 cross‑sections. Then it is necessary to analyse whether the remaining effective cross‑section is able to absorb the internal forces. On the other hand, Simplex Method performs a plastic calculation of stresses, which is not affected by the c/t ratios of the cross‑section parts. Therefore, there is no analysis of local buckling, but only a plastic stress analysis.
SHAPE-THIN performs classification of thin‑walled cross‑sections according to EC 3 and EC 9 by determining effective widths as well as effective cross‑section properties. Stresses can be checked accordingly with regard to a reduced cross‑section. In contrast, the plastic analysis according to Simplex Method does not consider any local buckling effects. This may lead to more favourable design ratios. However, these are not achieved in the cross‑section because of stability failure.
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Figure 01 - Setting Section from Library
Figure 02 - Axial Force and Bending Moments
Figure 03 - SHAPE-THIN Table '6.2 Classification of the Cross-Section According to EN 1993-1' and Stress Diagram
Figure 04 - Internal Forces of Member No. 2: Axial Force Without Moment My
Figure 05 - Stresses on Full and Reduced Cross-Section (Class 4)
Figure 06 - Von Mises Yield Criterion (Simplex Method)
Figure 07 - SHAPE-THIN Table '8.1 Plasticity' and Plastic Equivalent Stresses for Member No. 2 (Class 4)