*When analysing the stability of a steel framework, it is important to make the difference between sway and non-sway frames.*

*A frame may be classified as non-sway if its response to in-plane horizontal forces is sufficiently stiff for it to be acceptably accurate to neglect any additional internal forces or moments arising from horizontal displacements of its nodes. For a given load case, portal frames with shallow roof slopes and beam-and-column type plane frames may be classified as non-sway if the elastic critical load ratio αcr for that load case satisfies the criterion:*

*Figure 1. Effects of deformed geometry of the structure - Notations*

In this case the internal forces and moments may be determined using the initial geometry of the structure (first-order analysis).

Any other frame shall be classified as a sway frame and the effects of the horizontal displacements of its nodes (PΔ effects) taken into account in its design.

If each portal frame of the structure is considered to deform independently, then a 2D modelling approach is preferred as it allows for a quite straightforward computation of the elastic critical load ratio. The first step of this computation is to determine which load case or load combination can generate elastic instability of the structure. In other words, we need to identify the load case/combination producing the highest vertical loads on the bottom of the structure.

For the 2D portal frame depicted below, wind and snow loads are automatically generated in Advance Design, Structural Analysis and Design software. With these and the self-weight of the structure, we can generate load combinations according to EN 1990.

*Figure 2. Structure assumptions and geometry - 2D*

*(Click image to enlarge)*

After performing a finite element analysis of the structure, we issue the table containing the sum of reactions on supports and look for the load case or load combination giving the highest vertical reaction on supports. In this case, combinations 138 to 140 give a value of 59.85 kN. These combinations are related to the same wind load case and to an identical value of snow load, but have been differentiated to account for three different snow scenarios. As the instability generated on the structure by these three combinations is the same, we can continue working with combination 138 (1.35 x [1G] + 1.5 x [17 Snw] + 0.75 x [13 WY-D]).

*Figure 3. Sum of reactions on supports – 2D model*

*(Click image to enlarge)*

The second step is to predict the Euler’s critical load at which buckling can occur on the portal frame. In Advance Design this is made possible by the “Generalised buckling” analysis. This analysis gives a lambda coefficient (λ) and the distribution of efforts for each buckling eigen mode. Lambda is the ratio of the buckling loads to the currently applied loads, or in other words the factor by which the design loading would have to be increased to cause elastic instability in a global mode. This is equivalent to the elastic critical load ratio αcr from the Eurocodes.

To perform the “Generalised buckling” analysis, create a new calculation assumption from the Project browser by right-clicking on “Analysis settings”. Select “Generalised buckling” from the context menu and choose the load combination for the analysis from the dialog box.

*Figure 4. Generalised buckling analysis - Creation*

*Figure 5. Generalised buckling analysis - Load combination*

The new analysis appears now in the Project browser. It contains a load case named “Buckling”. Input the number of vibration modes from the properties table of this load case (see below). As the elastic critical load ratio is usually provided by the first eigen mode of the structure, which is a global mode, we can choose to perform this analysis for a small number of modes, for instance three.

*Figure 6. Generalised buckling analysis - Number of modes*

We perform a new finite element analysis of the structure. Once the calculation sequence is finished, we access the “Analysis and Combinations” from the “Results settings” dialog box. Under the “Buckling” load case we can visualise the elastic critical load ratio of each buckling mode. By displaying each mode in the graphic area, we can identify the one presenting a global in-plane deformation of the structure. In this case it is Mode 1 with an elastic critical load ratio of 45.098. As this value is greater than 10, the portal frame can be classified as non-sway.

*Figure 8. Results settings - Analysis and combinations*

*(Click image to enlarge)*

*Figure 9. Elastic critical load ratio - Portal frame in 2D*

*(Click image to enlarge)*

The stability analysis is also possible in a 3D environment, even if in this case it will be more difficult to identify the buckling mode producing the global deformation of the structure. It is very unlikely that this would be the first eigen mode, which is why we will need to analyse a greater number of modes. In order to avoid having too many local modes, we will choose not to mesh the purlins and side rails of the framework.

*Figure 10. Structure assumptions and geometry - 3D*

*(Click image to enlarge)*

From the “Structure settings” dialog box we select the 3D modelling mode, and to ensure consistency with the 2D model, we consider the same assumptions with regards to the climatic loading. After creating load combinations according to the Eurocodes, we perform the finite element analysis of the structure.

From the table with the sum of reactions on supports, we identify the combinations 144 to 146 as the ones giving the highest vertical reaction on supports, which is 247.991 kN. As for the 2D model, these three combinations are related to the same wind load case and to an identical value of snow load. Therefore, we will choose to continue working with combination 144 (1.35 x [1G] + 1.5 x [18 Snw] + 0.75 x [15 WY-D]).

*Figure 11. Sum of reactions on supports - 3D model*

*(Click image to enlarge)*

We create a “Generalised buckling” analysis in the Project browser, and after selecting the load combination for the analysis we ask the software to generate 100 vibration modes.

Once the results are available, we graphically display each buckling mode until we identify the one generating the global deformation of the structure. In our case the buckling mode 58 presents a global deformed shape very similar to the one obtained with the 2D model. For this mode, the elastic critical load ratio is 43.37, which is also very close to the value obtained in 2D.

*Figure 12. Elastic critical load ratio - 3D framework - Front view*

*(Click image to enlarge)*

*Figure 13. Elastic critical load ratio - 3D framework - Perspective view*

*(Click image to enlarge)*

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**References:***1992 – National Application Document - ENV 1993-1-1 Eurocode 3: Design of steel structures - Part 1.1: General rules and rules for buildings – European Committee for Standardization2000 – The Institution of Structural Engineers – Manual for the design of steelwork building structures to EC3 – ISBN 1-874266-53-02001 – Manfred Hirt, Michel Crisinel – Traité de Génie Civil de l’École polytechnique fédérale de Lausanne – Charpentes métalliques – Conception et dimensionnement des halles et bâtiments – Presses polytechniques et universitaires romandes - Volume 11 – ISBN 2-88074-359-12005 – EN 1993-1-1 Design of steel structures – Part 1-1: General rules and rules for buildings - European Committee for Standardization*