compressive axial forces act along the respective chords (see Fig. 4(c) once more) — therefore, the P– moment distributions are nonlinear and local (i.e., involve each member inpidually).

  (iii) Obviously, the P–  effects only exist in unbraced frames (in braced frames, all member chords remain unchanged).In spite of the above description and although this paper deals exclusively with P–  effects, it is important to stress the fact that their exact values can only be determined if the P– effects are properly incorporated in the analysis. However,one should also mention the existence of several approximate methods to estimate P–  effects that either fully neglect or consider only partially the P– effects—provided that certain requirements are fulfilled, such methods have been found to yield rather accurate results.

3.2. Elastic design approaches

   In general, the design or safety checking of an unbraced plane frame, with respect to ultimate limit states involving inplane buckling phenomena, must account for both the P–δand P–Δsecond-order effects — concerning the incorporation of these geometrically non linear effects in an elastic design or safety checking procedure, it is worth mentioning that:

  (i) According to the current steel design codes, the P–δeffects must always be taken into account. As for the P–Δeffects, one is only forced to include them whenever the vertical loading acting on the frame is “sufficiently high”. In the particular case of EC3-EN (2005), this concept is quantified by means of the factor cr = Fcr /FEd , where FEd and Fcr are the load parameter values defining the(i1) frame applied vertical loading and (i2) critical loading associated with elastic buckling in a global sway mode:the P–Δ effects must be taken into consideration if cr <10. In order to avoid the need to perform a frame linear stability analyses (to obtain Fcr ), EC3-EN proposes the use of the approximate expression (i is the number of frame storeys)

   which is just a slight modification of the well-known Horne’s method and, thus, exhibits the distinctive advantage of involving only the results of an elastic first-order global analysis of the frame. However, it is explicitly stated that this method is only applicable to the sway mode buckling of (i1) orthogonal beamand-column and (i2) shallow-roof (  < 26— seeFig. 1) portal (i.e. pitched-roof) frames,‘provided that the axial compression in the rafters is not significant’. Because a considerable number of pitched-roof frames commonly used in practice (e.g. in industrial buildings) are acted by quite high rafter compressive axial forces, the above condition automatically precludes the application of Eq. (8) to those frames.

  (ii) Regardless of the cr value, the P–  and P–effects may always be included simultaneously in the frame internal force and moment (IFM) design values, which are then used to check the (elastic or elastic–plastic) cross-section resistance of its members. However, this implies that these IFM values must be obtained through rigorous elastic second-order global analyses, which must incorporate all the relevant frame and member imperfections.

  (iii) If cr   10, it suffices to perform a first-order global analysis of the frame — the internal IFM values obtained are then used to check the overall resistance of each inpidual frame member, by means of beam–column interaction formulae that have supposedly been calibrated to take into account all of the relevant member buckling phenomena and imperfections.

  (iv) If cr < 10, it is enough to incorporate the P–  and P–effects separately in the design or safety checking procedure. In the first stage, the P–Δ effects are included in the frame IFM design values, a task that can be achieved either directly or indirectly and requires the performance of a global analysis that needs to include only the relevant frame imperfections. Then, in the second stage, these IFM design values are used to check the resistance of each inpidual frame member, as described in the previous item.