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where 2keLr is the buckling length of the fictitious member and ke accounts for the characteristics of the frame columns and support conditions. For instance, it is possible to calculate S by means of the ‘nonsway’ chart included in Annex E of EC3-ENV.
The 36 unbraced pitched-roof frames analysed exhibit commonly used configurations, defined by combinations of the following geometric parameters: = 6, 12; RL = Lr /Lc = 2, 3, 4; RI = Ic/Ir = 1, 2; K = 0, 1,1. The set of RN ( Nr /Nc) values considered corresponds to column and rafter axial forces due to a transverse load uniformly distributed along the frame span — thus, these RN values depend on RI , RL , Sand .3 Table 2 shows values of RN and the ASB and SB ratios
ap/ ex concerning the various frames and associated with the analytical formulae proposed by (i) the authors (ASB and SB— Eq. (1) and Table 1), (ii) Davies (ASB — Eq. (2)) and (iii) Trahair (SB — Eq. (3)). The analysis of the numerical results presented in this table lead to the following conclusions:
(i) The RN values vary between 0.59 ( = 6, RI = 1, RL =2, K = 0) and 2.19 ( = 6, RI = 2, RL = 4, K = 1) and, for 21 of the 36 frames analyzed, one has RN > 1
(higher rafter compression).
(ii) The proposed formulae yield ( ap/ ex )AS values ranging between 0.93 and 1.02. The results are conservative for 28 frames (average error of 4%), exact for 4 frames and non conservative for the remaining 4 frames (all fixed-base).
(iii) Davies’s formula yields ( ap/ ex )AS values (for K = 0 and K = 1) varying from 0.83 to 1.04. The estimates are conservative for 20 frames (average error of 12%) and non conservative for the remaining 4 frames (all fixed-base). In general, the pinned-base frame estimates are less accurate.
(iv) The proposed formulae yield ( ap/ ex )S values ranging between 0.95 and 1.04. The results are conservative 14 frames (average error of 2.5%), exact for 8 frames and non conservative for the remaining 14 frames (average error of
2.4%);
(v) Trahair’s formula yields ( ap/ ex )S values varying from 0.38 to 0.93 (average error of 32%, but all values conservative). Due to the nature of the formula, the accuracy of its estimates decreases as and RN increase — i.e. as the rafter relative stiffness diminishes).
In order to provide a better understanding of the accuracy and limitations of the formulae developed by Davies and Trahair, one addresses next the qualitative nature of the results yielded by them. First of all, it is convenient to rewrite the three formulae in the format of Eq. (1), which requires taking C = 1. Concerning the two Davies’s formulae, they can be cast in the form:
where one has respectively for K = 1 and K = 0 — it is interesting to note that the c.0 and r.0 expressions appearing in Eq.(6) are identical to the ones given in Table 1 (for K = 0).Eq. (4) clearly shows that the approximation proposed by
Davies amounts to replacing the frame exact ‘stability curve’by the straight line segments shown in Fig. 3(a1) (pinnedbase frames) and 3(a2) (fixed-base frames) — they unite the points (Nc.0, 0) and (0, Nr.0), defined by Eqs. (5) and (6). One observes that:
(i) In pinned-base frames (Fig. 3(a1)), the straight line segment provides a reasonably accurate approximation of the exact curve and the estimates obtained are almost always conservative (except in the vicinity of the end points).
(ii) In fixed-base frames (Fig. 3(a2)), it is obviously not possible to approximate the exact curve by a single straight line segment with reasonable accuracy — note that it exhibits a fairly elliptic shape (C = 2). Then, it is not surprising that the straight line segment associated with Davies’s formula only approximates accurately the stability curve ‘upper portion’, corresponding to large RN values — indeed, the approximation of the curve ‘lower portion’ is poor.