(iv) Estimation of the second order displacement and IFM values, by means of the expressions:

   Concerning the above sway amplification method, the following comments are appropriate:

  (i) Unlike its antisymmetrical counterpart NAS, the axial force component NSS is not amplified — see Eq. (17)4. This stems from the fact that the signs of the equal-valued additional bending moments generated by the P–  effects at the column bases are the same in the first case and opposite in the second one (see Fig. 9(a)–(b)). Therefore,no additional ‘axial force binary’ N  is required to ensure equilibrium in the presence of the (M )SS —they are selfequilibrated.

  (ii) If one uses the analytical expressions given in Eq. (1) and Table 1 to evaluate FAS and FS, it is possible to obtain second-order displacement and IFM estimates which (ii1) include all relevant P–   effects and (ii2) are exclusively based on the results of a frame first-order global analysis.

  (iii) Unlike the sway amplification method prescribed by EC3-ENV and EC3-EN, which was presented earlier (see Eqs. (9)–(11)), the proposed method involves the amplification of two first-order displacement and IFM components (instead of just one). The differences between the second-order estimates yielded by the two methods depend (iii1) on the values of the ratios FS/FAS and MSS. max/MAS.max , and (iii2) on whether the symmetrical sway component (not addressed in EC3, most likely because it bears no relevance in orthogonal beam-andcolumn frames) is included in the nonsway component, which is not amplified, or in the antisymmetrical sway component, which is amplified by means of CAS.

4.4. Validation — parametric study

   In order to investigate the accuracy and range of validity of the proposed sway amplification method, an extensive parametric study was performed, involving a set of 33 pitchedroof frames with E = 210 GPa and distinct geometrical and loading characteristics (see Fig. 6(a)–(b)). Initially, a set of 27 frames was constructed by (i) considering a reference frame having   = 10, Lc = 5 m, IPE360 columns, FEd/FAS = 0.25 and HEd/FEd = 0.10, and (ii) varying the frame span (L = 20, 30, 40 m), the rafters inertia (IPE300, IPE360, IPE450) and the column base stiffness (kc = 0, 6800 kN m,1).For each of these frames, after performing a linear stability analysis to determine the ratio FS/FAS, a comparison was made between exact and approximate second-order (i) horizontal displacements and bending moments at node D (MII , dII),

(ii) rafter shear forces at node D (VII) and (iii) rafter CD and column DE axial forces (NI I.r , NI I.c) — see Fig. 6(b).While the exact values were determined through geometrically nonlinear analyses carried out in the commercial code ABAQUS, the approximate ones were obtained by means of the proposed SAM. For comparison purposes, one also presents bending moment estimates yielded by the SAM prescribed by both the EC3-ENV and EC3-EN, which were evaluated by adopting the following two approaches: including MSS (not covered in EC3) in either (i) the nonsway component MN S (not amplified) or (ii) the anti-symmetrical sway component MAS (amplified by CAS). The corresponding second-order moments are given by:

  All the results concerning the analysis of the aforementioned 27 frames are presented in Table 3. In addition, (i) the results shown in Table 4 make it possible to assess the influence of the rafters slope   (reference frames with L = 30 m, IPE360 rafters and kc = 6800 kN m and four   values), and (ii) Table 5 deals with frames acted by vertical loads alone, i.e., such that HEd = 0 (reference frames with L = 20 m, IPE300 rafters and three kc values). The results of these parametric comparative studies prompt the following conclusions and/or remarks:

  (i) The ratio FS/FAS varies from 2.02 to 3.88 (pinnedbase frames), 1.05 to 1.39 (fixed-base frames) and 1.15 to 1.92 (semi-rigid frames). As for the ratio   =MSS. max/MAS. max, it ranges from 4.2 to 7.3 (pinned-base frames), 12.5 to 32.6 (fixed-base frames) and 6.9 to 20.6(semi-rigid frames).