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2. Buckling behaviour of pitched-roof frames
Consider the symmetrical single-bay unbraced pitched-roof frame depicted in Fig. 1(a), with semi-rigid column bases (stiffness kc) and subjected only to column (Nc) and rafter (Nr )axial compressive forces, also symmetrical and related by the
parameter RN = Nr /Nc. It has been shown that the in-plane global buckling behaviour of this frame is governed by the two buckling modes shown in Fig. 1(b), with antisymmetrical (ASM) and symmetrical (SM) configurations — both involve
sway displacements, i.e. horizontal displacements at the column tops2 .
Fig. 2 summarizes the essential features of the pitchedframe elastic buckling behaviour, which were unveiled after the performance of a number of parametric studies, involving fixed,semi-rigid and pinned-base frames [17,22]. In particular, it is worth noting that the value of the critical load parameter cr =min{ AS; S}, where AS and S are the load parameter values associated with antisymmetrical and symmetrical buckling(ASB and SB), is strongly influenced by both (i) the rafter slope and (ii) the axial load ratio RN —this is clearly shown by the curves depicted in Fig. 2. These curves provide the bifurcation values of Nc and Nr (both depend linearly on ) and N ASc.0 , NSc.0,N ASr.0 and NSr.0 are bifurcation loads associated with RN = 0(column axial forces only) and RN = 1 (rafter axial forces only). Their observation prompts the following remarks:
(i) While the frame ASB behaviour is independent of the rafter inclination (solid line), its SB counterpart exhibits a marked dependence on (dashed lines) — indeed, it is mostly governed by the rafter instability.
(ii) For low RN values, cr = AS (no dependence on ). For high RN values, on the other hand, one has cr = AS or cr = S, depending on the value of the rafter inclination .
(iii) The variation of cr with RN involves either (iii1) boththe AS and S buckling modes (e.g., = 1) or (iii2) only the ASM (e.g., = 3). Moreover, it is worth noting that frames with commonly used geometries often exhibit quite close AS and S values.
2.1. Bifurcation load formulae
In order to avoid the need to perform linear stability analyses to estimate AS and S, easy-to-use analytical expressions have been developed by the authors, in the context of frames with semi-rigid column bases. The bifurcation loads are yielded by the expressions:
where c.0 and r.0 are given by the formulae included in Table 1, in which (i) NEc and NEr are the column and rafter Euler loads, (ii) Nc and Nr are the column and rafter reference forces and (iii) one has R = Lr Ic/Lc Ir , RH = Lr sin /Lc and K = kcLc/E Ic. Note that, by making K tend to either zero or infinity, one obtains the pinned and fixed-base frame formulae. It is worth noting that these formulae have been included in a recent book by Trahair et al on the design and behaviour
of steel structures.
2.2. Accuracy of the proposed formulae and comparison with alternative ones
In order to assess the accuracy and range of validity of the proposed formulae, one presents next the results of a parametric study involving a fair number of unbraced pitched-roof frames with different geometries and column base stiffness values. It is possible to compare the values of AS and S yielded by (i) exact linear stability analyses, (ii) the proposed analytical formulae (see Eq. (1) and Table 1) and (iii) the analytical formulae developed by Davies and Trahair .Davies developed analytical formulae aimed at estimating the values of AS in pitched-roof frames with pinned (K = 0) and fixed (K = 1) column bases. They read
On the other hand, Trahair proposed the estimation of S through the critical buckling load of a fictitious uniformly compressed member with length 2Lr , which corresponds to the length of two horizontal rafters ( = 0 ) — this buckling load is given by the expression,