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3.2.1. Incorporation of the P–Δeffects in the IFM design values
When dealing with frames having cr < 10 (the socalled sway frames), the most common design or safety checking approach consists of addressing the P–Δ and P–δeffects separately. Then, it is essential to incorporate the P–Δeffects in the frame IFM design values, a task that may be performed either (i) directly, through exact or approximate genuine second-order global analyses or provided that certain requirements are met, (ii) indirectly,by means of an appropriate modification of the results yielded by first-order global analyses. The most widespread (and rational) indirect methods available are the so-called ‘sway
amplification methods’(SAM), a designation stemming from the fact that they are based on the identification and proper amplification of the first-order ‘sway moments’ MS and the‘sway displacements’ dS. According to a SAM, the design values of the frame moments and displacements (MEd and dEd )are given by:
where (i) MN S and dN S are the ‘nonsway’ bending moments and displacements (note that the first-order moments MI and displacements dI are the sum of their sway and nonsway‘components’4), and (ii) C is an amplification factor. Both the EC3-ENV (1992) and EC3-EN (2005) prescribe a SAM based on the amplification factor
and the latter also states explicitly that this method is applicable to (i) single storey beam-and-column or shallow-roof portal (pitched-roof) frames, again provided that the beam/rafter axial compression is not significant, and to (ii) multi-storey frames with similar vertical load, horizontal load and shear stiffness distributions per storey. In addition, EC3-EN further stipulates that this amplification method can only be applied to framessuch that cr 3.05 — it is worth noting that this limitation is not the same in other codes prescribing the same SAM (e.g.EC3-ENV).
As bending moments are the only IMF covered by the overwhelming majority of SAM descriptions (e.g. the one appearing in EC3-ENV), one must also address the way in which the axial and shear forces must be handled, in order to end up with a set of second-order IMF satisfying equilibrium and compatibility as much as possible. In orthogonal beam-andcolumn frames, the second-order axial and shear force values are determined by means of a procedure involving the following steps:
(i) Use of Eqs. (9) and (10) to determine the second-order bending moments and displacements.
(ii) By considering the global equilibrium at the frame deformed configuration, determine the column axial forces on the basis of the second-order moments and displacements obtained in (i). This is illustrated in Fig. 5(a)–(b), for a simple frame acted by horizontal loads: while Fig. 5(a) shows the frame first-order deformed configuration (equal to its sway component) and IMF,Fig. 5(b) addresses the equilibrium of the additional bending moments M , due to the applied vertical loads P and sway displacements — since both M have the same direction, equilibrium requires the development of self-equilibrated additional column axial forces N .Therefore, the second-order axial forces are given by summing P (first-order nonsway), N (first-order sway) and N (P–Δ effects) — note that (N + N ) may be viewed as ‘amplified sway axial forces’.
(iii) By considering the equilibrium at the deformed configuration of each column, determine the column shear forces on the basis of the moments, displacements and axial forces calculated in the previous two steps. Note that, since these values are approximate, the sums of the column shear forces in the storeys do not fully equilibrate the corresponding applied horizontal load resultants。
(iv) By considering the equilibrium at each beam-to-column node, determine the beam axial and shear forces on the basis of their column counterparts.