From an eigenvalue analysis, KAB = 3.69 is obtained. It is seen that a direct use of the alignment chart leads to a significant error for this frame, and other approaches give good results. However, the LeMessurier approach requires the use of the alignment chart, and the Lui approach requires a firstorder analysis subjected to a fictitious lateral loading.

31.7.2 Leaning Columns

Recognizing that a leaning column is being braced by rigid columns, a model for the leaning column as shown in Figure 31.15b was proposed by Lui (1992). Rigid columns provide lateral stability to the whole structure and are represented by a translation spring with a spring stiffness SK. The K-factor for a leaning column can be obtained as

In2 EI


For most commonly framed structures, the term (h2£I/SkL3) normally does not exceed unity, and so K = 1 often governs. AISC (1999) suggests that leaning columns with K = 1 may be used in unbraced frames provided that the lack of lateral stiffness from simple connections to the frame (K =1) is included in the design of moment frame columns. Aristizabal-Ochoa (1994b) recommended that (1) the K-factors of leaning columns are identical to the K-factors of the rigid columns when they are subjected to the same magnitude axial loads and are made of the same section and (2) the K-factors of leaning columns must be greater than 1.0 or the K-factor corresponding to the fully braced column with the same supports or boundary conditions.

31.7.3 Remarks

Numerical studies by Geschwindner (1995, 2002) found that the Yura approach gives overly conservative results for some conditions, the Lim and McNamara approach provides sufficiently accurate results for design, and the LeMessurier approach is the most accurate, among the three. The Lim and McNamara approach could be appropriate for preliminary design, while the LeMessurier and Lui approaches would be appropriate for final design.

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