Mechanics and effective lengths of columns with positive and negative end restraints

In approximate buckling analysis of frame structures, it is common practice to simplify by considering compression members in isolation with end restraints that supposedly reflect the interaction with the surrounding structure. Buckling modes and effective lengths of such members have traditionally been limited to modes that can be associated with positive end restraints. A wider range of realistic buckling modes, that also imply negative end restraints, are reviewed and discussed, and approximate formulas for effective lengths and inflection point locations are developed from simple physical models for unbraced and fully braced compression members. The formulas are expressed in terms of convenient end fixity factors that allow for simple and elegant formulations. The range of applicability of the formulas is identified by comparisons with exact results for a reasonably wide range of positive and negative restraints, and applications to special structures are demonstrated.