The procedure just described readily shows that, according to both the EC3-ENV and EC3-EN, all the IMF design values of an unbraced sway orthogonal beam-and-column frame(NEd   NII , VEd   VII , MEd   MII) can be obtained,from the associated first-order values, by means of the common expression

   where (i) C is given by Eq. (10) and (ii) cr (i.e., Fcr ) is calculated on the basis of the first-order axial force profile due to the applied vertical loading.6 This explains why EC3-EN,unlike its ‘predecessor’ EC3-ENV, states explicitly that ‘For single storey frames designed on the basis of elastic global analysis, second order sway effects due to vertical loads may be calculated by increasing the horizontal loads and equivalent loads due to imperfections and other possible sway effects according to first order theory by the factor (1 − 1/cr )−1.’

4. Second-order analysis of pitched-roof frames

   Let us consider the pitched-roof frame shown in Fig. 6(a),which is identical to the one depicted in Fig. 1(a) and is now subjected to the vertical (H) and horizontal (w =FEd/L) loads indicated in Fig. 6(b). Before addressing the geometrically nonlinear behaviour of this frame, it is important to recall some fundamental differences between the structural behaviours of pitched-roof and orthogonal (beam-and-column) frames :

  (i) In orthogonal frames, the way in which the (nodal) applied horizontal loads are distributed among the two column tops does not affect the corresponding sway displacements —they are always virtually identical. In pitched-roof frames,

on the other hand, these displacements are only equal if the two horizontal applied loads are identical (due to the sloping rafters lateral stiffness).

  (ii) Symmetrical (geometry and vertical loading) orthogonal frames exhibit null displacements at the column tops. In vertically loaded symmetrical pitched-roof frames, there are no negligible ‘outward’ column sway displacements, which stem from the combined effect of the vertical displacement of the frame apex and the rafter inclination.

  (iii) It is a well-known fact that the first-order sway deformed configuration and bending moment distribution of a symmetrical orthogonal frame are exclusively due to the applied horizontal loads — this feature is very important, as the validity of the SAM is based on the similarity between these first-order results and their frame critical buckling mode counterparts [11]. In the next subsection,it will be shown that pitched-roof frames do not share this feature.

4.1. First-order deformed configuration and bending momentdistribution

   It has been shown that it is possible to express the first-order deformed configuration and bending moment distribution of a pitched-roof frame acted on by the loads shown in Fig. 6(b) as the sum of the following three components, shown in Fig. 7(a)–(c):

  (i) The non sway components dN S and MN S, due to the vertical loading and assuming that the sway displacements are prevented by the horizontal reaction forces R — see Fig. 7(a).

  (ii) The symmetrical sway components dSS and MSS, which stem also from the vertical loading, through the application of horizontal forces opposite to the reactions R — see Fig. 7(b).

  (iii) The anti-symmetrical sway components dAS and MAS, due to the horizontal loads7 — see Fig. 7(c).

   It is still worth mentioning that the maximum MSS values are very often considerably larger than the corresponding MAS ones — indeed, the two charts presented in Fig. 8(a) (fixedbase frames) and 8(b) (pinned-base frames) provide values that lead to a straightforward evaluation of the ratio between the maximum MSS and MAS values: for any given geometry (RL = Lr /Lc, RH = RL sin  , R = IcLr /Lc Ir ) and loading (FEd = wL, HEd = 2H) combination, the determination of   = MSS. max/MAS. max just requires the multiplication of  0 or  1, (read from the appropriate chart) by the factor (FEd/HEd )(RL cos  /18). In frames with semi-rigid column bases (K = kcLc/E Ic),   can be approximately estimated by means of