(iii) It may be shown that Davies’s fixed-base formula yields an exact result for RN = 4/(RI R2L )   R  N — value corresponding to the intersection with the exact curve (see Fig. 3(a2)). Therefore, it is possible to state that this formula provides fairly accurate and almost always conservative estimates if RN   R N . Note that none of the frames analyzed significantly violates this condition (see Table 2)—recall that the RN values are due to a uniformly distributed transverse load.

  (iv) For 0   RN < R N , Davies’s fixed-base formula always provides non conservative estimates, which become progressively more inaccurate (the straight line is always‘outside’ the exact curve). Frames exhibiting low RN values are fairly common in industrial buildings with crane girders — the axial compression is much higher in the columns. 

   Trahair’s formula can also be cast in the form of Eq. (1) (with C = 1), leading to:

and corresponding to the horizontal straight line Nc = 0 shown in Fig. 3(b), which is tangent to exact curve associated with   =0 at the point defined by (0, Nr.0) — recall that this formula was developed for an orthogonal beam-and-column frame.Thus, the estimates provided by Trahair’s formula are often excessively conservative — for a given RN value, the error increases with   (horizontal line more ‘inside’ the exact curve).Indeed, these estimates might give the (wrong) impression that  S is practically always the frame critical buckling load. Finally, note that Trahair’s formula only yields nonconservative estimates in frames with quite low RN values — none of the frames analysed here fulfils this condition (see Table 2).

3. Second-order analysis and design fundamentals

 3.1. Second-order effects—identification and characterisation

   Second-order effects in frames are often designated P–delta effects, due of the fact that they stem essentially from additional moments generated by (i) vertical applied forces and/or (ii) member compressive axial forces acting on the frame deformed configuration. The P–delta effects are (i) displacements, mainly related to serviceability limit states, and (ii) internal forces and moments, mostly concerning the ultimate limit states — the bending moments are always particularly relevant. In the literature, it is usual to make a distinction between two types of P–delta effects, namely (i)the P–  effects and (ii) the P–Δeffects. These two types of second-order effects are illustrated in Fig. 4, for the case of a simple orthogonal beam-and-column frame, and can be characterized as follows:

  (i) The P–  effects are due to the vertical forces applied to the frame and concern only its sway displacements  ,i.e. the relative displacements taking place between the column ends (see Fig. 4(b))—the deformations occurring inside the member spans are not taken into account.Moreover, the additional bending moments associated with these second-order effects (P–Δmoments) are obtained from equilibrium equations established in a frame‘partially deformed’ configuration, defined by the chords of its members (see again Fig. 4(b)) — this automatically implies that the P–Δmoment distribution is linear and global (i.e. involves the whole frame).

  (ii) The P–effects are due to the compressive axial forces acting on the various frame members and concern their inpidual deformations, i.e. the displacements that take place between the member deformed configurations and chord positions (see Fig. 4(c)). As for the additional bending moments associated with these second-order effects (P– moments), they are yielded by equilibrium equations written in the deformed configurations of the various members and assuming that the corresponding