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Next, the comparison between the frame (i) first-order sway deformed configurations dSS and dAS, shown in Fig. 7(b) and (c), and (ii) symmetrical (S) and antisymmetrical (AS) buckling mode shapes, displayed in Fig. 1(b) and (c) shows a remarkable similarity. This observation, combined with the likelihood of the (more or less) close proximity between the corresponding buckling loads FAS and FS (total vertical loadings associated with AS and S), leads to conclusion that,in pitched-roof frames, an approximate second-order analysis based on the amplification concept must involve the first and second buckling loads/modes — this fact constitutes a major difference with respect to the orthogonal beam-and-column frames, in which only the first buckling load/mode is involved.
4.2. Displacement and moment amplification
Generally speaking and provided that some requirements are met, the second-order deformed configuration of a frame acted an arbitrary single-parameter ( ) proportional loading may be carried out approximately, by solving the following sequence of problems :
(i) Determination of the member axial force distribution (firstorder analysis).
(ii) On the basis of the axial force profile obtained in the previous step, determination of the frame bifurcation loads (defined by the parameter value sequence 1b< 2b< · · · < nb< · · ·) and the shapes of the corresponding (normalised) buckling modes (designated as d1, d2, . . . , dn, . . .).
(iii) Determination of the frame first-order deformed configuration (essentially due to member bending), which is designated as dI .
(iv) Decomposition of dI into a linear combination of terms involving the frame (normalised) buckling modes, thus yielding and determination of the coefficients q j I (modal analysis).
(v) Determination of an estimate of the frame second-order deformed configuration (dII) that is associated with a given applied load level (defined by ), by means of the expression:
(vi) Since the bending moments are proportional to the curvatures, one may also estimate the frame bending moment distribution associated with dII , which is yielded by:
where M j is the bending moment distribution associated with the buckling mode d j .
Therefore, each component of the first-order deformed configuration dI = qj Idj and bending mome distribution MI = qjI Mj is amplified through an amplification factor given by:
which involves the corresponding bifurcation load parameter jb . Finally, it is worth noting that (i) the amplification expressions (14) and (15) only remain valid if the frame deformed and undeformed configurations are not too far apart (this is nearly always the case in practical applications) and that (ii) the second-order shear force and axial force distributions can be readily determined on the basis of equilibrium considerations — see Section 3.2.1.
4.3. Sway amplification method for pitched-roof frames
The concepts presented earlier are now used to formulate a sway amplification method specifically aimed at estimating the second-order (i) lateral displacements and (ii) elastic internal forces and moments in unbraced symmetrical pitched-roof
frames. It involves the following steps:
(i) Determination of the first-order member axial forces due to the applied vertical load only.
(ii) Evaluation of the frame sway mode buckling loads FAS = AS FEd and FS = S FEd .
(iii) Identification of the first-order deformed configuration,bending moment distribution, shear force distribution and axial force distribution components associated with (iii1) the application of the vertical load in the braced frame,
(iii2) the sway displacements due to the vertical load and (iii3) the sway displacements due to the horizontal loads(see Fig. 7(a)–(c)).